Unfiltered FQPSK: Another Interpretation and Further Enhancements

that occur along this pair of paths, we have
2 Ts
/ 2
2
2
dmin = ∫ [( s t s t s t s t
Ts
/ 3() − ()) + ( () − ())
− 2
3
4 5
(14)
2
2
+ ( s () t −s () t s () t s () t
2 14 ) + ( −
0 3 )
2
2
+ s() t −s () t s () t s () t ] dt
Evaluation of the squared Euclidean distances
between the pairs of waveforms
required in (14) using (7a) and (7b) for their
definition results after much algebra in
d
( ) + ( − )
(15)
The average signal (I+Q) energy is obtained
from
1 15 15
Ts
/ 2 2
2
Eav
= ∑ ∑ ∫ [ si
( t) + s t dt
Ts
i
( )] =
− / 2
256 i=
0 j=
0
⎡
Ts
⎤
Ts
2 1 15
∑ ∫ si
t dt s t dt
Ts
Ts
i
⎣
⎢
i
⎦
⎥ = 1 7
/ 2 2
/ 2 2
()
− 2
∑ ∫ ()
/
− / 2
16 = 0
4 i=
0
which again using (7a) and (7b) evaluates to
E
(16)
(17)
Since the average signal (symbol) energy is twice the
average energy per bit, then the normalized minimum
squared Euclidean distance for the paths corresponding
to starting and ending in the same state is
2
d
2E
1 5
⎡7
8 ⎛ 3 4 ⎞
− − A +
4 3 ⎝ 2 3 ⎠ + ⎤
⎢ π π ⎥
= ⎢
⎥T
=
s
1 552Ts
⎢
2⎛11
4⎞
⎥
⎢ A +
⎣ ⎝ 4 π ⎠ ⎥
⎦
2
.
min
b
12 14
⎛
2
7+ 2A+
15A ⎞
= ⎜
⎟ T = 0.
9946 T
⎝ 16 ⎠
av s s
⎡
2
16 7 8 ⎛ 3 4 ⎞ 11 4
− − A + A
4 3 ⎝ 2 3 ⎠ + ⎛
4
+ ⎞⎤
⎢
=
⎣ π π ⎝ π⎠⎥
⎦
= 156
2
7+ 2A+
15A
min
.
( )
(18)
▲ Figure 11. The optimum trellis-coded receiver for FQPSK.
Upon examination of all length 3 error event paths
that begin in one state and end in another, e.g., Figures
10a and 10b, no pair of paths with smaller normalized
minimum squared Euclidean distance was found.
Furthermore, by exhaustive search, it can be shown
that the minimum squared Euclidean distance of (18) is
the smallest over all pairs of paths that start in any state
and end in any state regardless of the length of the path.
Thus, the normalized minimum squared Euclidean distance
for the FQPSK scheme is given by (18).
For the spectrally enhanced FQPSK using the waveforms
of (9) as replacements for their equivalents in
(7b), the minimum squared Euclidean distance over all
length 3 trellis paths occurs, for example, between the
first and second paths, starting and ending in state
“0000” and is given by (see Figure 9a)
2 Ts
/
dmin = ∫ [ s t s t s t s t
Ts
/ 0( ) −
1( ) + ( ) − ( )
− 2
0 12
2
2
+ s () t −s () t s () t s () t
+ s () t −s () t s () t s () t ] dt
(19)
Once again, evaluation of the squared Euclidean distances
between the pairs of waveforms required in (19)
using (7a) and (7b) together (now with (9) for their definition)
results after much algebra in
d
Likewise, the average signal energy is now
E
2 2
( ) ( )
( ) + ( − )
2
( )
0 3
0 4
2
( 0 2 ) + −
0 0
(20)
(21)
Thus, the normalized minimum squared Euclidean
distance is
d
2E
2
⎡3− 6A+
15A ⎤
= ⎢
T 1 564
s T
s
⎣ 4
⎥ =
⎦
2
min
.
⎛ 21 8 ⎛ 1 8 ⎞ 29
− − A⎜
− ⎟ + A
⎜ 8 3π
⎝ 4 3π⎠
8
⎜
4
⎜
⎝
3− 6A+
15A
= 156
8 ⎛ 1 8 ⎞ 29 2
− − A − A
3π ⎝ 4 3π⎠ + 8
2 2
min
=
.
b
21
8
( )
⎞
⎟
⎟T
⎟
⎠
= 1.
003T
av s s
2
2
(22)
90 · APPLIED MICROWAVE & WIRELESS