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

Performance Analysis of a JTIDS/Link-16-type
Waveform Transmitted over Nakagami Fading
Channels with Pulsed-Noise Interference
Abstract—The Joint Tactical Information Distribution System
(JTIDS) is the communication terminal of Link-16. JTIDS is a
hybrid direct sequence/frequency-hopping spread spectrum
system and features Reed-Solomon codes for channel coding,
cyclic code-shift keying for 32-ary symbol modulation, minimumshift
keying for chip modulation, symbol interleaving, chip
sequence scrambling and random jittering for transmission
security, and a double-pulse structure for diversity. Assuming
that coherent chip demodulation is practical, we investigate the
probability of symbol error of a JTIDS/Link-16-type waveform
for both the single- and the double-pulse structure transmitted
over a slow, flat Nakagami fading channel in the presence of
pulsed-noise interference (PNI) in this paper. In general, the
results show that the double-pulse structure always outperforms
the single-pulse structure, whether the PNI is present or not and
whether the channel is fading or not. Furthermore, barrage noise
interference has the most effect in degrading performance when
signal-to-interference ratio (SIR) is small. When SIR is large,
PNI with a smaller fraction of time that interference is on causes
the greatest degradation.
Index Terms—Joint Tactical Information Distribution System
(JTIDS), Link-16, Nakagami fading, pulsed-noise interference,
probability of symbol error.
I. INTRODUCTION
he Joint Tactical Information Distribution System (JTIDS)
Tis
the communication component of Link-16. JTIDS is a
hybrid direct sequence/frequency-hopping (DS/FH) spread
spectrum system and features Reed-Solomon (RS) codes for
channel coding, cyclic code-shift keying (CCSK) for 32-ary
baseband symbol modulation, minimum-shift keying (MSK)
for chip modulation, symbol interleaving, chip sequence
scrambling and random jittering for transmission security, and
a double-pulse structure for diversity. In this paper, a
JTIDS/Link-16-type system is first introduced, and then the
probability of symbol error of a JTIDS/Link-16-type
waveform for both the single- and the double-pulse structure
transmitted over a fading channel in the presence of pulsednoise
interference (PNI) is investigated. Given the
assumptions that the chip duration is less than the channel
coherence time and that at a particular hop the signal
bandwidth is less than the channel coherence bandwidth, the
fading channel is modeled as a slow, flat Nakagami fading
978-1-4244-2677-5/08/$25.00 ©2008 IEEE
Chi-Han Kao, Frank Kragh, and Clark Robertson
Electrical and Computer Engineering Department
Naval Postgraduate School, Monterey, CA 93943-5121
channel. The performance of hybrid DS/FH spread spectrum
systems for various modulation schemes in the presence of
different types of interference and fading has been
investigated in [1]-[5], but only [1] attempts to evaluate the
performance of the JTIDS waveform analytically in AWGN.
Unfortunately, the results presented in [1] are based on the
overly optimistic assumption that the cross-correlation values
for the CCSK symbol demodulator are independent. Actually,
it can be shown that these cross-correlation values are not
independent.
Currently, the JTIDS waveform is received noncoherently
at the chip level, but in this paper the performance of a JTIDStype
waveform with coherent detection is evaluated in order to
ascertain the performance possible if coherent chip
demodulation were practical. The analysis presented in this
paper can easily be modified to evaluate performance with
noncoherent chip demodulation.
II. SYSTEM MODEL DESCRIPTION
A. JTIDS-type Transmitter
The system model of a JTIDS-type transmitter is shown in
Figure 1. As is seen, the transmitter consists of a (31,15) RS
encoder, a symbol interleaver, a CCSK 32-ary baseband
symbol modulator, a data chip scrambler, a frequency-hopping
circuit, a MSK chip modulator, and a transmitting antenna.
Link-16
messag
(31,15)
RS Encoder
Symbol
Interleaver
CCSK
M-ary Symbol
Modulation
Fig. 1 Model of a JTIDS-type transmitter.
32-Chip
PN Sequence
Scrambing
Frequency
Hopping
MSK
Chip
Modulation
JTIDS
Pulses
When the Link-16 message bit stream arrives at JTIDS, it is
first mapped into 5-bit symbols. The seven symbols of the
message header are encoded with a (16,7) block code that is
related to a RS code by shortening and/or puncturing. The 15
symbols of the message data are encoded with a (31,15) RS
code. After encoding, the data and header symbols are
interleaved for the first layer of transmission security. Next,
these 5-bit interleaved symbols are modulated with 32-ary
CCSK, where each 5-bit symbol is represented by one of the
cyclic-shifts of a 32-chip starting sequence. To obtain the

second layer of transmission security, each 32-chip CCSK
sequence is scrambled with a 32-chip pseudo-noise sequence.
The resulting 32-chip sequence is modulated for transmission
with MSK to generate analog pulses. Each pulse is then upconverted
to one of the 51 possible carrier frequencies, which
contributes a third layer of transmission security. Normally,
the starting point of the pulse train is pseudo-randomly
jittered, which provides a fourth layer of transmission security
since it can make it difficult for a jammer to decide when to
turn on the jamming signal. After up-conversion, the signal is
amplified, filtered, and transmitted over the channel [6].
B. JTIDS-type Receiver
At the receiver (Figure 2), the receiving process is the
reverse of the transmission process. After frequency dehopping,
MSK chip demodulation, and de-scrambling by the
32-chip PN sequence, each 5-bit channel symbol is recovered
by a CCSK symbol demodulator. The determination of which
5-bit channel symbol was received is done by computing the
cross-correlation between the de-scrambled 32-chip sequence
and all possible 32 sequences, and the decision is made by
choosing the 5-bit channel symbol corresponding to the
branch with the largest cross-correlation value. After symbol
de-interleaving, the channel data symbols are decoded by a
(31,15) RS decoder. If the decoding is successful, the data
symbols are converted into a bit stream which are sent to the
upper layer.
Link-16
message
(31,15)
RS Decoder
Symbol
De-interleaver
CCSK
32-ary Symbol
Demodulator
Fig. 2. Model of a JTIDS-type receiver.
32-Chip
PN Sequence
De-scrambling
MSK
Chip
Demodulator
Frequency
De-hopping
III. PERFORMANCE ANALYSIS
Given the assumptions that frequency de-hopping is
perfectly synchronized with the frequency-hopped waveform
and that the signal-to-noise ratio (SNR) is large, the MSK chip
demodulator recovers the original scrambled 32-chip
sequence. Given that de-scrambling is perfectly synchronized,
the CCSK symbol demodulator detects the original 5-bit
coded symbol. As seen in Figure 2, to evaluate the probability
of symbol error of a JTIDS/Link-16-type waveform, the
probability of channel chip error at the output of the MSK chip
demodulator and the probability of channel symbol error at the
output of the CCSK symbol demodulator are both required.
A. Probability of Channel Chip Error
MSK can be considered as a special case of offset
quadrature phase-shift keying (OQPSK) with sinusoidal pulse
shaping [7]. When a coherent matched filter or correlator is
used to recover the data chips, MSK has the same performance
as binary PSK and OQPSK; that is,
⎛
Pc= Q ⎜
⎝
2E
⎞ c
⎟,
N ⎟
0 ⎠
(1)
where E c is the average energy per chip. Since each 5-bit
symbol is converted into 32 chips, E = 5E = 32E
; where
s b c
E S is the average energy per symbol and E b is the average
energy per bit. Therefore, (1) can be expressed as
⎛ 10E
⎞ b
Pc= Q⎜
⎟.
(2)
⎜ 32N
⎟
⎝ 0 ⎠
The JTIDS waveform is currently received noncoherently at
the chip level, but the performance with coherent detection is
evaluated in order to determine performance if coherent chip
demodulation were practical. The analysis presented in this
paper can easily be modified to evaluate noncoherent chip
demodulation. From (2), when forward error correction (FEC)
coding is used, the probability of channel chip error for a
JTIDS/Link-16-type waveform in AWGN is given by
⎛ 10E ⎞
b ⎛ 10rE
⎞
c
b
pc= Q⎜ ⎟=
Q⎜
⎟,
(3)
⎜ 32N ⎟ ⎜
0 32N
⎟
⎝ ⎠ ⎝ 0 ⎠
where Eb = rE
c b is the average energy per coded bit, and r is
the code rate.
The Link-16 message data can be sent with either a singlepulse
structure or a double-pulse structure [8]. The doublepulse
structure increases the anti-jam capability of the link
since it provides a diversity of L = 2 . Note that (3) is only
valid for the single-pulse structure. To recover the data sent
with the double-pulse structure, soft decision (SD) combining
on a chip-by-chip basis is assumed in the coherent MSK chip
demodulator as shown in Figure 3.
rt ()
πt
cos cos 2π
ft c
2T
c
c
( 2k 1)
Tc
( 2k 1)
T dt
+
−
∫
( 2k+ 2)
Tc
∫ dt
2kTc
πt
sin sin 2π
fct 2T
t = ( 2k+ 2) Tc
c
t = ( 2k+ 1) Tc
1 st pulse
even chips
2 nd pulse
even chips
1 st pulse
odd chips
2 nd pulse
odd chips
SD combining
+
Σ
+
+
Σ
+
SD combining
Fig. 3. MSK coherent chip demodulator with double-pulse/SD combining.
When the double-pulse structure is used, JTIDS is a hybrid
DS/FFH spread spectrum system with sequential diversity
L = 2 since each symbol is transmitted twice on two different
carrier frequencies. Thus, the average energy per symbol is
Es = LEp
(4)
where L = 2 , and E p is the average energy per pulse. Since
Es = 5Eb
and E p = 5Eb',
from (4) we get
Eb = LEb'
(5)
where E b ' is the average energy per bit per pulse. Using (5) in
(3), we obtain an expression for the probability of channel
chip error of a JTIDS/Link-16-type waveform in AWGN as
+
-
+
-
S/P
ˆS

⎛ 10rLE
⎞ b'
pc= Q⎜
⎟,
(6)
⎜ 32N
⎟
⎝ 0 ⎠
where L = 1 for the single-pulse structure, and L = 2 for the
double-pulse structure. Note that if p c is expressed in terms
of b E instead of LE b'
, then p c is the same for both the singleand
the double-pulse structure in AWGN.
B. Probability of Channel Symbol Error
The probability of symbol error for the 32-ary CCSK
sequence chosen for JTIDS is given by [9]
32
Ps= ∑ P{ symbol error N = j} P{ N = j}
(7)
j = 0
where N is a binomial random variable which represents the
total number of chip errors occur at the output of the MSK
chip demodulator. Since the demodulation of CCSK is
independent of the FEC coding, the probability of channel
symbol error can be obtained from (7) as
32 ⎛32⎞ j
32−
j
ps = ∑ ζ j ⎜ ⎟ pc ( 1−
pc)
(8)
j=
0 ⎝ j ⎠
where ζ j are the conditional probabilities of channel symbol
error given that N = j chip errors occur at the output of the
MSK chip demodulator, and p c is the probability of channel
chip error. In [9], the values of ζ j were obtained both
analytically and by Monte Carlo simulation. Since the analytic
result ζ UB yields a tight upper bound, it is used to evaluate
j
JTIDS’s performance in this paper. Therefore, from (8),
32 ⎛32⎞ j
32−
j
ps < ∑ ζ UB p ( 1 ) .
j ⎜ ⎟ c − pc
(9)
j = 0 ⎝ j ⎠
C. Performance Analysis in AWGN
As mentioned earlier, JTIDS uses a RS code for FEC
coding. A RS code is a linear, nonbinary block code. For a tsymbol
error correcting, nonbinary block code, the probability
of decoder, or block, error is upper bounded by [10]
n ⎛n⎞ i
n−i PE ≤ ∑ ⎜ ⎟ ps( 1−
ps)
(10)
i=+ t 1⎝i⎠
where the equality holds for either a perfect code or a bounded
distance decoder, and p s is the probability of channel symbol
error. Since each block consists of n coded symbols, the
probability of symbol error for a linear, nonbinary block code
is obtained by approximating the probability of information
symbol error given i channel symbol errors per block by in
[11]. Substituting this into (10), we get
n 1 ⎛n⎞ i
n−i PS ≈ ∑ i⎜ ⎟ ps( 1 − ps)
. (11)
n i=+ t 1 ⎝i⎠ Now, using (6) with r = 15 31 and either L = 1 or 2 in (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 either the single- or the
double-pulse structure over AWGN.
D. Performance Analysis in both AWGN and PNI
When a JTIDS/Link-16-type waveform is subjected to both
AWGN and PNI, (11) can still be used to evaluate the
probability of symbol error since (11) is independent of the
types of noise and/or fading channels. Since the probability of
channel symbol error is determined at the symbol level instead
of the chip level and since the PNI is assumed, the probability
of channel symbol error shown in (8) must be modified as
L ⎛L⎞
L−
ps = ∑⎜ ⎟ ρ1 ( 1−ρ1)
ps
(12)
=
0 ⎝⎠ where L = 1 for a single-pulse, L = 2 for a double-pulse, and
0< ρ1
≤ 1 represents the fraction of time the PNI is turned on.
Note that ρ 1 = 1 represents barrage noise interference (BNI).
The probability of channel symbol error given that pulses
are jammed s p is upper-bounded by
32 ⎛32⎞ 32−
j
j
ps < ∑ζ
UB p ( 1 ) ,
j ⎜ ⎟ c − pc
(13)
j=
0 ⎝ j ⎠
where = 0,..., L . The probability of channel chip error given
that pulses are jammed c p is given by
⎛ 0.3125rLE ⎞
b '
pc= Q⎜
⎟.
(14)
⎜ N0 + ( NILρ1) ⎟
⎝
⎠
Now, using (12) through (14) in (11) with r = 15 31,
and
either L = 1 or 2, we obtain the probability of symbol error of
a JTIDS/Link-16-type waveform for either the single- or the
double-pulse structure in both AWGN and PNI.
E. Performance Analysis over Nakagami Fading Channels
The process of evaluating the performance of a
JTIDS/Link-16-type waveform for the single-pulse structure
in both AWGN and PNI over Nakagami fading channels is
similar to that for channels with no fading. The only difference
is that (14) with L = 1 is now a conditional probability of
channel chip error since the received signal amplitude a c
fluctuates and is modeled as a Nakagami random variable with
a probability density function (pdf)
m
2
2 ⎛ m ⎞ ⎛
2m−1 −ma
⎞
c
ACc ⎜ ⎟
( ) 2
c ⎜ ⎟
2
c
Γ m ⎜a ⎟ ⎜
c a ⎟
c
f ( a ) = a exp , a ≥0
(15)
⎝ ⎠ ⎝ ⎠
where Γ() i is the Gamma function, and m is the fading
figure. When m < 1 , the fading is more severe than Rayleigh
fading; m = 1 is Rayleigh fading; when m > 1 , there is a lineof-sight
(LOS) component to the received signal, and when
2
m →∞, there is no fading. Replacing E b ' with aT c b'
in (14)
with L = 1 and = 0 , we get
⎛ 2
0.3125racT
⎞
b'
pc ( a )
,
0 c = Q⎜
⎟
(16)
⎜ N ⎟
⎝ 0 ⎠
where T b ' is the bit duration per pulse. Similarly, with L = 1
and = 1 , we get

⎛ 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.

P s
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
Single-pulse
Double-pulse
10
0 2 4 6 8 10 12 14 16 18 20
-9
E /N (dB)
b′ o
Fig. 4. Probability of symbol error (tight upper bound) for a JTIDS/Link-16type
waveform in AWGN: single-pulse versus double-pulse structure.
When PNI is present, the probabilities of symbol error for
both the single- and the double-pulse structure are shown in
E N = 15 and 10 dB, respectively.
Figures 5 and 6 where b'
0
Several observations can be made. First, for both the singleand
the double-pulse structure, the value of ρ 1 that maximizes
the probability of symbol error decreases as Eb'N I increases.
For example, for the double-pulse structure in Figure 5, ρ 1 = 1
has the most effect in degrading performance when Eb'N I is
less than 2 dB, while ρ 1 = 0.1 causes the greatest degradation
when Eb'N I is greater than 7 dB. Similar results are seen in
Figure 6. Second, the double-pulse structure outperforms the
single-pulse structure, whether ρ 1 is large or small. For
example, in Figure 5, for ρ 1 = 0.5 , the double-pulse structure
−5
outperforms the single-pulse structure by 4.2 dB at PS
= 10 ,
and for ρ 1 = 0.1,
the double-pulse structure outperforms the
−5
single-pulse structure by 3.2 dB at PS
= 10 . Lastly, the
double-pulse structure outperforms the single-pulse structure
by a greater margin as Eb ' N 0 decreases. For example, in
Figure 6 where Eb ' N 0 is 10 dB, for ρ 1 = 0.5 , the doublepulse
structure outperforms the single-pulse structure by 5.2
−5
dB (an increase of 1 dB) at PS
= 10 , and for ρ 1 = 0.1,
the
double-pulse structure outperforms the single-pulse structure
by 3.5 dB (an increase of 0.3 dB).
Since JTIDS is operated in the UHF band (LOS is required),
the range of the fading figure is 1 < m > 1,
the
performance for both the single- and the double-pulse
structure when the signal is transmitted over a slow, flat
Nakagami fading channel is virtually identical to that obtained
when there is no channel fading as shown in Figures 5 and 6.
When m = 2 , the probabilities of symbol error of a
JTIDS/Link-16-type waveform for both the single-pulse
structure (in both AWGN and PNI) and the double-pulse
structure (in both AWGN and BNI) are shown in Figures 7
E N = and 10 dB, respectively.
and 8 where ' 0 15
b
P s
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
ρ 1 = 1, SP
ρ 1 = 0.5, SP
ρ 1 = 0.3, SP
ρ 1 = 0.2, SP
ρ 1 = 0.1, SP
ρ 1 = 1, DP
ρ 1 = 0.5, DP
ρ 1 = 0.3, DP
ρ 1 = 0.2, DP
ρ 1 = 0.1, DP
10
0 2 4 6 8 10 12 14 16 18 20 22 24
-9
E /N (dB)
b′ I
Fig. 5. Probability of symbol error for a JTIDS/Link-16-type waveform in
E N = dB: single- versus double-pulse structure.
AWGN and PNI where ' 0 15
b
P s
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
ρ 1 = 1, SP
ρ 1 = 0.5, SP
ρ 1 = 0.3, SP
ρ 1 = 0.2, SP
ρ 1 = 0.1, SP
ρ 1 = 1, DP
ρ 1 = 0.5, DP
ρ 1 = 0.3, DP
ρ 1 = 0.2, DP
ρ 1 = 0.1, DP
10
0 2 4 6 8 10 12 14 16 18 20 22 24
-9
E /N (dB)
b′ I
Fig. 6. Probability of symbol error for a JTIDS/Link-16-type waveform in
E N = dB: single- versus double-pulse structure.
AWGN and PNI where ' 0 10
b
P s
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
ρ 1 = 1, SP
ρ 1 = 0.5, SP
ρ 1 = 0.3, SP
ρ 1 = 0.2, SP
ρ 1 = 0.1, SP
ρ 1 = 1, DP
10
0 2 4 6 8 10 12 14 16 18 20 22 24
-9
E /N (dB)
b′ I
Fig. 7. Probability of symbol error of a JTIDS/Link-16-type waveform for the
single-pulse structure (in both AWGN and PNI) and the double-pulse
structure (in both AWGN and BNI) transmitted over a slow, flat Nakagami
fading channel where ' 0 15
b
E N = dB and m =
2.

P s
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
ρ 1 = 1, SP
ρ 1 = 0.5, SP
ρ 1 = 0.3, SP
ρ 1 = 0.2, SP
ρ 1 = 0.1, SP
ρ 1 = 1, DP
10
0 2 4 6 8 10 12 14 16 18 20 22 24
-9
E /N (dB)
b′ I
Fig. 8. Probability of symbol error of a JTIDS/Link-16-type waveform for the
single-pulse structure (in both AWGN and PNI) and the double-pulse
structure (in both AWGN and BNI) transmitted over a slow, flat Nakagami
fading channel where ' 0 10
b
E N = dB and m = 2.
From Figures 7 and 8, several observations can be made
when compared to Figures 5 and 6, respectively. First, as
expected, similar results but poorer performances are observed
for smaller m. Second, for a fixed Eb ' N 0 , the double-pulse
structure outperforms the single-pulse structure by a greater
margin for smaller m . For example, in Figure 7 where m = 2 ,
for ρ 1 = 1 , the double-pulse structure outperforms the single-
−5
pulse structure by 4.1 dB at PS
= 10 , while in Figure 5
where m →∞, for ρ 1 = 1 , the double-pulse structure
−5
outperforms the single-pulse structure by 3.4 dB at PS
= 10 .
Lastly, the double-pulse structure outperforms the single-pulse
E N decreases. For
structure by a greater margin as b ' 0
example, in Figure 7 where b'
0
E N = 15 dB, the double-pulse
structure outperforms the single-pulse structure by 4.1 dB for
ρ 1 = 1 at PS
−5
= 10 , while in Figure 8 where Eb ' N 0 is
reduced to 10 dB, the double-pulse structure outperforms the
single-pulse structure by 6.5 dB for ρ 1 = 1 at PS
−5
= 10 .
V. CONCLUSION
In this paper, the probability of symbol error of a
JTIDS/Link-16-type waveform for the single-pulse structure
(in both AWGN and PNI) and the double-pulse structure (in
both AWGN and BNI) transmitted over a slow, flat Nakagami
fading channel is investigated. Several conclusions can be
drawn as follows. First, the double-pulse structure always
outperforms the single-pulse structure whether the PNI is
present or not and whether the channel is fading or not.
Second, for the single-pulse structure, the value of ρ 1 that
maximizes the probability of symbol error decreases as
E N increases; that is, BNI has the most effect in
b'I degrading performance when b'I E N is small. When Eb'N I
is large, PNI with a smaller fraction of time that interference is
on causes the greatest degradation, whether the channel is
fading or not. Third, when PNI is present in channels with no
fading, the double-pulse structure always outperforms the
single-pulse structure, whether b ' 0
E N is large or small.
Fourth, when the JTIDS signal is transmitted over a slow, flat
Nakagami fading channel, the double-pulse structure
outperforms the single-pulse structure by a greater margin for
a smaller value of m when b ' 0
E N is fixed. Lastly, for a
fixed value of ρ 1 , the double-pulse structure outperforms the
single-pulse structure by a greater margin for decreasing
E N , whether the channel is fading or not.
b ' 0
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