Performance Analysis of a JTIDS/Link-16-type Waveform ...

⎛ 0.3125ra
T ⎞
pc ( a )
.
1 c = Q
⎜ ⎟
2
c b'
⎜ ⎟
⎝
N0 + NIρ1 ⎠
2
If γ = aT N , (16) can be rewritten as
b c b'
0
( γ ) ( γ )
c0b b
(17)
p = Q 0.3125 r ,
(18)
and the pdf in terms of γ b is given by
m
m−1
γ ⎛ b m ⎞ ⎛−mγ ⎞ b
fΓ
( γ )
exp , 0
b b = ⎜ ⎟ ⎜ ⎟ γb
≥ (19)
Γ( m)
⎜γ ⎟ ⎜
b γ ⎟
⎝ ⎠ ⎝ b ⎠
2
where γ b = aT c b'
N0
is defined as the ratio of the average
energy per bit per pulse-to-noise power spectral density. If
( )
γT 2
= aT c b' ⎡⎣N0 + NI
ρ1
⎤⎦
, we can rewrite (17) as
p γ = Q 0.3125 rγ
,
(20)
( ) ( )
c1T T
and the pdf in terms of γ T is given by
m
m−1
γ ⎛ T m ⎞ ⎛−mγ ⎞ T
fΓ
( γ )
exp , 0
T T = ⎜ ⎟ ⎜ ⎟ γT
≥
Γ( m)
⎜γ ⎟ ⎜
T γ ⎟
⎝ ⎠ ⎝ T ⎠
2
where γ = aT N + ( N ρ )
(21)
T c b' ⎡⎣ 0 I 1 ⎤⎦
. The average probability of
channel chip error when the PNI is off is obtained from
pc0 ∞
= ∫ pc ( γ ) ( ) .
0 b fΓγ b b dγ
−∞
b
(22)
Substituting (18) and (19) into (22), we obtain the average
probability of channel chip error when the PNI is off as
m
m−1
∞ γ ⎛ b m ⎞ ⎛−mγ ⎞ b
c = ( γ ) 0 0
b ⎜ γb
( m)
⎜ ⎟
γ ⎟ ⎜ ⎟
b γ ⎟
b
p ∫ Q 0.3125r Γ ⎝ ⎠
exp
⎝
d
⎠
. (23)
When m is an integer, (23) can be evaluated to obtain [12]
m m−1
k
⎛1− μ ⎞ ⎛m− 1+
k⎞⎛1+
μ ⎞
pc
=
0 ⎜ ⎟ ∑ ⎜ ⎟
2 k = 0 k
⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝
2 ⎠
(24)
μ = γ m + γ and γ0= 0.3125rγ 2 . Similarly,
where 0 ( 0)
the average probability of channel chip error when the PNI is
on is given by
m m−1
k
⎛1− ν ⎞ ⎛m− 1+
k⎞⎛1+
ν ⎞
pc
=
1 ⎜ ⎟ ∑ ⎜ ⎟
2 k = 0 k
⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝
2 ⎠
(25)
ν = γ m + γ and γ1= 0.3125rγ 2 . Using (24)
where 1 ( 1)
and (25) in (13) along with ζ UB from [9], we obtain
j
T
b
p s and
0
p s , respectively. Using p
1
s and p
0
s in (12), we obtain p
1
s .
Using p s in (11), we obtain the probability of symbol error of
a JTIDS/Link-16-type waveform for the single-pulse structure
in both AWGN and PNI when the signal is transmitted over a
slow, flat Nakagami fading channel.
For the double-pulse structure, it is difficult to investigate
the performance for a JTIDS/Link-16-type waveform in both
AWGN and PNI when the signal is transmitted over
Nakagami fading channels since it is extremely complex to
obtain an analytic expression for the average probability of
channel chip error given that one pulse is jammed. Instead, the
performance for a JTIDS/Link-16-type waveform in both
AWGN and BNI transmitted over a slow, flat Nakagami
fading channel is evaluated. In this case, we assume maximalratio
detection with linear combining. For maximal-ratio
detection with linear combining when both AWGN and BNI
are present, the conditional probability of channel chip error
given that both pulses are affected by BNI is
⎛ 2 2
0.3125r(
ac + ac ) T ⎞
b'
pc( ac) = Q⎜
⎟.
(26)
⎜ N0+ N ⎟
I
⎝ ⎠
We can rewrite (26) as
p γ = Q 0.3125r
γ + γ
(27)
( )
c( T ) ( T T )
since
2
aT ( N N ) γ γ γ
γ T = c b' 0 + I . If T T T
∗ ∗
( γ ) ( γ )
∗ = + , (27) becomes
p = Q 0.3125 r ,
(28)
c T T
and the pdf in terms of γ T
∗ is given by
f
( )
( )
∗
2m−1 2m
γ ∗
T ⎛ m ⎞ ⎛ ∗ −mγ
⎞ T ∗
T ⎜ ⎟ T
γ ⎟ ⎜ ⎟
T γ ⎟
T
γ = exp , γ ≥0
Γ ⎝ ⎠ ⎝ ⎠
∗
ΓT
( 2m)
where
2
aT ( N N )
(29)
γ T = c b' 0 + I .
The average probability of channel chip error given that
both pulses are affected by BNI is
∞
∗ ∗ ∗
pc = pc ( γT ) f ∗ ∫ ( γT ) dγT.
−∞
ΓT
Substituting (28) and (29) into (30), we obtain
(30)
( )
∗
2m−1 2m
γ ∗
∞
T ⎛m⎞ ⎛ ∗ −mγ
⎞ T ∗
c = ( γ ) 0
T ⎜ γT
( 2m)
⎜ ⎟
γ ⎟ ⎜ ⎟
T γ ⎟
T
p ∫ Q 0.3125r Γ ⎝ ⎠
exp
⎝
d
⎠
. (31)
When m is an integer, (31) can be evaluated to obtain
2m 2m−1 k
⎛1− β ⎞ ⎛2m− 1+
k⎞⎛1+
β ⎞
pc
= ⎜ ⎟ ∑ ⎜ ⎟
2 k = 0 k
⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝
2 ⎠
(32)
β γ 2m γ γ = rγ
. Using (32) in
= + and 2 0.3125 T
where 2 ( 2)
(9) along with ζ UB from [9], we obtain
j
s p . Next, using p s in
(11), we obtain the probability of symbol error of a
JTIDS/Link-16-type waveform for the double-pulse structure
in both AWGN and BNI when the signal is transmitted over a
slow, flat Nakagami fading channel.
IV. NUMERICAL RESULTS
The probabilities of symbol error of a JTIDS/Link-16-type
waveform for both the single- and the double-pulse structure
in AWGN are shown in Figure 4. As expected, the doublepulse
structure outperforms the single-pulse structure in terms
of average energy per bit per pulse b '
b ' 0
while the b ' 0
E . At
P
−5
S = 10 , the
E N required for the double-pulse structure is about 4 dB,
E N required for the single-pulse structure is
about 7.1 dB. In other words, the double-pulse structure
−5
outperforms the single-pulse structure by 3.1 dB at PS
= 10
in AWGN.