Anti-jamming and Anti-multipath Performances of Generalized FH/BFSK

Anti-jamming and Anti-multipath Performances of
Generalized FH/BFSK
Yi-Chen Chen
Department of Electrical Engineering
University of Washington, Seattle, WA 98195, USA
[email protected]
Abstract - We consider and investigate a generalized signaling
method for non-coherent orthogonal FH/BFSK in band multitone
jamming (BMJ). We let symbol tones not be restricted within one
common band, but be pre-designed or randomly selected from all
available tones. We present hopping assignments that are more
resistant to jamming and multipath fading. Simulation results
show that in n = 1 BMJ and AWGN, performances of all assignments have similar asymptotic behaviors in high signal-tojammer-ratio (SJR) conditions, whereas in n = 2 BMJ and
AWGN, there exists around 6 dB performance gain in the proposed assignments over the conventional FH/BFSK when the bit
error probability (BEP) level is 10 -2 . In 2-path Rayleigh fading
channel corrupted by n = 1 BMJ, the performance gain is at least
15 dB. The orthogonality between two contiguous symbol tones in
conventional FH/BFSK is severely destroyed by the delayed second path. This kind of multipath interference, however, can be
greatly alleviated by the generalized signaling presented in this
paper.
Kwang-Cheng Chen
Department of Electrical Engineering
National Taiwan University, Taipei 106, Taiwan
[email protected]
ing
jammed
(later,
we
denote
this
probability
by
p ( m | nvec, qvec) .) by simply putting m jamming tones in
some selected FH/BFSK bands, and further optimizes its jamming power to maximize the bit error probability (BEP) of the
desired user’s communication.
The conventional non-coherent FH/BFSK also possesses a
disadvantage regarding its vulnerability to the multipath interference [10]. In multipath environments, even if the transmission is conducted in the noise-free or fading-free channel, the
orthogonality between two contiguous BFSK symbol tones is
vanished, since the original (first) path is corrupted by other
randomly delayed paths. The resulting performance degradation cannot be neglected because adjacency is always maintained for both symbol tones in every hopping interval, except
that we can re-design the FH/BFSK in a more generalized way.
Band (1)
Band (2) Band (3) Band (4)
I. INTRODUCTION
Frequency hopping (FH) technique is widely used in military, consumer and commercial applications nowadays to effectively combat various kinds of interference such as multipath fading, multiple access interference and interference from
co-existence systems. In FH communications, hostile jamming
remains an issue of critical interests, and is a dominating factor
to affect the performance degradation. Among several jamming strategies in previous research, band multitone jamming
(BMJ) is shown to be one of the most powerful jamming
strategies [1-9].
In conventional non-coherent FH/BFSK, each FH/BFSK
modulation band consists of two adjacent and equally spaced
symbol tones. In BMJ [1], the jammer is assumed to have the
ability to exploit the structure of modulation bands. All jamming tones are pseudo-randomly distributed over the spread
spectrum bandwidth such that there are n0 jamming tones in
each jammed FH/BFSK bands, where n0 = 1 or 2. This strategy is called n = n0 BMJ [1]. In Fig. 1, the modulation band
consists of Tone (3) and Tone (4). The jamming tones are located in Tone (4) and Tone (8). Note that in each hopping interval, Tone (2i – 1) and Tone (2i) for some i (i = 1, 2, 3, or 4)
always form a modulation band. With this knowledge of
modulation band structure, the jammer controls the conditional
probability of exact m symbol tones in a modulation band be-
1-4244-0063-5/06/$20.00 ©2006 IEEE
1
2
3
4
5
6
7
8
Fig. 1: Conventional FH/BFSK in n = 1 BMJ
Band (1)
1st hopping interval
2nd hopping interval
Band (2) Band (3) Band (4)
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Fig. 2: One kind of the generalized FH/BFSK signaling
In this paper, we consider a generalized signaling method
for non-coherent FH/BFSK. We here let both symbol tones not
be restricted within one common band, but be pre-designed or
randomly selected from all available tones. Fig. 2 shows one
kind of generalized FH/BFSK signaling. Note that Tone (3)
and Tone (8) form a legitimate FH/BFSK modulation band in
the first hopping interval. The modulation band in the second
hopping interval is changed to be Tone (1) and Tone (5), and
so on. The jammer cannot exploit the desired user’s unfixed
modulation band structure in this generalized signaling format.
In addition, it allows more flexibility (these two symbol tones
can possibly be separated far from each other) to alleviate the
non-orthogonal interference caused by the multipath.
We present simulation results to show hopping assignments
that are more resistant to jamming and multipath fading. We
also conduct experiments to verify reasons for unsatisfactory
performances of the conventional FH/BFSK under two-path
Rayleigh fading. In APPENDIX, we analyze asymptotic behaviors in high SJR conditions for each assignment under n =
2 BMJ and explain the inherent performance gain of the proposed assignments.
II. SYSTEM MODEL
We consider a slow frequency hopping (SFH) system with
non-coherent orthogonal binary frequency shift keying (BFSK)
in the presence of band multitone jamming (BMJ) and additive
white Gaussian noise (AWGN). Under the control of a hopping sequence, the carrier frequency of the transmitted signal
hops in a pseudo-random manner over the entire spread spectrum band, Wss . We assume the timing recovery is ideally
done at the desired user’s receiver. The jammer does not have
knowledge of the desired user’s hopping sequences.
Let Rb and Eb respectively be the symbol rate and symbol
energy of the desired user. In this paper, we focus on the scenario of one-hop-per-symbol SFH with single diversity. The
symbol rate is equal to the hopping chip rate. The entire spread
spectrum band is partitioned into N t = Wss / Rb tones. These
tones are equally spaced and further partitioned into
Nb = Nt / 2 contiguous, non-overlapping FH/BFSK conventionally defined (CD) bands. S is the power of the transmitted
signal. J is the total jamming power. Q is the total number of
jamming tones each of which has power J / Q = S / α . This
definition follows (2.31) of 2.2.2.1 in [1]. N J ≡ J / Wss is the
power spectrum density of the equivalent jamming noise. The
transmitted signal tone in the l-th symbol interval (l = 0, 1, ...)
is expressed by Re[ Sl (t )] , where
Sl (t ) = 2 S p (t − lTs ) exp ª¬ j ( 2π f (l )t + ϕ s (l ) ) º¼ .
(1)
In (1), p (t ) is the normalized rectangular pulse function
with duration Ts = 1/ Rs . f (l ) is resultant of the desired user’s
hopping frequency plus the symbol’s carrier frequency in the
l-th symbol interval. The multitone jamming signal in the l-th
symbol interval is expressed by Re[ J l (t )] , where
Q
(
)
J l (t ) = ¦ q =1 2 S α p (t − lTs ) exp ª¬ j 2π f q (l )t + ϕ q (l ) º¼ . (2)
In (2), f q (l ) is the carrier frequency of the q-th jamming
tone in the desired user’s l-th symbol interval. ϕs (l ) and
ϕ q (l ) in (1) and (2) are random phases uniformly distributed
in [0, 2π ) . One symbol interval of each jamming signal synchronizes with one symbol interval of the desired user. In each
symbol interval, Q jamming signals are located in Q different
tones, which are randomly chosen from Nt tones.
II-A. Two Channel Types
In this paper, we consider two channel types: Type I channel: AWGN only, and Type II channel: AWGN and two-path
Rayleigh fading.
In BMJ and Type I channel, the received signal in the desired
user’s
l-th
symbol
interval
is
Rl (t ) = Re[ Sl (t ) + J l (t )] + n(t ) , where n(t ) is AWGN with
zero mean and two sided power spectral density N 0 / 2 .
In Type II channel, the impulse response of two-path
2
Rayleigh fading is h(t ) = ¦ βnδ (t − τ n ) exp( jθ n ) , where τ n
n =1
(n > 1) is uniformly distributed in [0, Tb ) . The n-th path amplitude β n is Rayleigh distributed with parameter β n2 2 . β n2
is the average power of the n-th path (i.e. mean square value of
the n-th path amplitude) and is expressed by
β n2 = β12 exp( −τ n / Tb ) . The corresponding phase difference
of the n-th path, θ n , is statistically independent of β n . It is
also assumed that τ 1 = 0 and β12 = 1 . In BMJ and Type II
channel, the received signal in the l-th symbol interval is
Rl (t ) = Re[( Sl (t ) + J l (t )) ∗ h(t )] + n(t ) .
At the receiver, Rl (t ) is de-hopped and then enters the noncoherent BFSK energy detector. The energy is calculated as
the sum of squared in-phase and quadrature phase components
at the output of correlators with normalized (i.e. unit-energy)
correlation basis functions.
II-B. Modulation Band Assignment Types
Fig. 3 depicts the structure of N b FH/BFSK CD bands and
the corresponding tones. In the following analysis, we let the
symbol tones: Tone ( t1 ), Tone ( t2 ), form a modulation band in
each hopping interval and comply with the requirement that t1 ,
t2 ∈ {1, 2, ..., Nt } and they are different. Note that in the generalized FH/BFSK, Tone ( t1 ), Tone ( t2 ) need not be located
in a common CD band.
We consider the following two types of modulation band
assignments.
1
2
3
Generally speaking, the generalized signal design’s complexity grows linearly with the symbol alphabet size.
Band ( N b )
Band (1) Band (2)
4
II-C. Two BMJ Types: n = 1 BMJ and n = 2 BMJ
( 2 N b − 1) 2 N b
Similarly, we can use “qvec = [q1 q2]” to describe the BMJ:
We let the Q jamming tones be located in Tone ( t1 ), Tone
( t 2 ), ..., and Tone ( t Q ). They comply with the requirement
Fig. 3: The structure of N b FH/BFSK CD bands and N t tones
II-B-1. Deterministic Type (“nvec = [n1 n2]”)
We use “nvec = [n1 n2]” to denote a tone assignment for the
modulation band in each hopping interval, where nk is defined
to be the total number of CD bands, in each of which there are
k symbol tones, and (2 – k) tones that are not symbol tones.
We define the FH/BFSK modulation band as following a
“nvec = [n1 n2]” assignment if its both symbol tones satisfies
the following equation in each hopping interval:
¦
Nb
i =1
δ «ª
¬
(¦
2
j =1
)
δ ¬ª «¬t j 2»¼ + 1 − i ¼º − k º» = nk ,
¼
k = 1, 2, (3)
where δ [x ] = 1 if x = 0 and δ [x] = 0 if x ≠ 0 . ¬x ¼ denotes
the largest integer ≤ x . In other words, both symbol tones in
any hopping interval are located over Nt tones such that exact
k symbol tones are located in a common CD band and there
are nk such CD bands, where nk is a non-negative integer.
2
“nvec = [n1 n2]” satisfies ¦ k ⋅ nk = 2 . In Fig. 1, the gener-
that t1 , t 2 , ..., t Q ∈ {1, 2, ..., N t } and are all different in each
hopping interval. We use “qvec = [q1 q2]” to denote a jamming-tone assignment, where qk is defined to be the total number of CD bands, in each of which there are k jamming tones
and (2 – k) tones that are not jamming tones. The Q jamming
tones follow “qvec = [q1 q2]” assignment if they satisfy the
following equation in each hopping interval:
¦
Nb
i =1
δ ª«
¬
(¦
Q
j =1
)
δ ª¬ «¬t j 2 »¼ + 1 − i º¼ − k »º = qk ,
¼
k = 1, 2. (5)
In other words, these Q jamming tones in any hopping interval are located over N t tones such that exact k of Q jamming tones are located in a common CD band and there are qk
such CD bands, where qk is a non-negative integer. “qvec = [q1
2
q2]” satisfies ¦ k ⋅ qk = Q . By definition, it follows that in
k =1
n = 1 BMJ, qvec = [Q 0], whereas in n = 2 BMJ, qvec = [0
Q/2].
k =1
alized FH/BFSK modulation band follows “nvec = [0 1]” assignment, as what the conventional FH/BFSK does. In Fig. 2,
it follows “nvec = [2 0]” assignment by keeping its two symbol tones located in different CD bands in each hopping interϬʳ
val.
II-B-2. Random Type (“nvec = random”)
III. SIMULATION RESULTS
We take N b = 200 as the total number of CD bands and
take Eb / N 0 = 30 dB for AWGN in both channel types. The
BMJ parameter setting follows strategies presented in page
482 – 491 of [1]:
1.
In n = 1 BMJ, the jammer sets α with Eb /(2 N J ) if
Eb / N J ≤ 2 . For Eb / N J > 2 , the jammer sets α with
0.95 so that each jamming tone’s power slightly exceeds S.
2.
In n = 2 BMJ, the jammer sets α with ( Eb / N J ) if
Eb / N J ≤ 1 , and with min[2.52, Eb / N J ] if
Eb / N J > 1 .
We define the FH/BFSK modulation band follows “nvec =
random” if p(Tone( t j ) = Tone(i) | E j ) = 1 ( N t − j + 1) for
∀i ∈ S j and j = 1, 2. We denote p(Tone( t j ) = Tone(i) | E j )
the conditional probability that Tone( t j ) equals Tone(i) given
the event E j that Tone ( t1 ), Tone ( t 2 ), ..., and Tone ( t j −1 )
have been selected from Nt tones. Subset S j is defined by
j −1
S j ≡ {1, 2, ..., N t } − * k =1{tk } .
(4)
III-A. BMJ and Type I Channel (AWGN only)
In other words, we randomly choose two distinct symbol
tones from N t tones to form a FH/BFSK modulation band in
each hopping interval.
Ϭ
The generalized scheme is more sophisticated because a
hopping pattern now consists of not just one CD band number
as in conventional FH/BFSK, but a set of numbers for both
tones. Two frequency synthesizers at the receiver are required
to respectively receive the signal at the two possible tones.
Fig. 4(a) and Fig. 4(b) respectively show simulation results
in terms of BEP versus Eb N J for generalized FH/BFSK
under n = 1 and n = 2 BMJ in Type I channel (AWGN only).
In n = 1 BMJ, three curves merge together in high SJR conditions, whereas in n = 2 BMJ, there is around 6 dB performance
gain over the conventional FH/BFSK (i.e. “nvec = [0 1]”)
when the BEP level is 10 −2 . Both Fig. 4(a) and Fig. 4(b) illus-
trate that “nvec = [2 0]” and “nvec = random” perform better
than the conventional FH/BFSK. In other words, both symbol
tones being located in different CD bands, or in a random
manner, results in more jamming resistance.
ments’ curves. This experiment helps reflect the fact that delayed paths are the main contributor to the severe performance
degradation of the conventional FH/BFSK (“nvec = [0 1]”), as
shown in Fig. 5.
Given nvec and qvec, the bit error probability (BEP) is expressed by
2
p (BEP | nvec, qvec) = ¦ m = 0 p(BEP|m) p (m | nvec, qvec) , (6)
where p (BEP | m ) is the conditional BEP given the event that
exact m symbol tones are jammed, and p (m | nvec, qvec ) is
the probability of this event. In APPENDIX, we investigate
the asymptotic behavior of p( m | nvec, qvec) in n = 2 BMJ, to
explain the performance gain inherent in the assignments:
“nvec = [2 0]” and “nvec = random”.
III-B. BMJ and Type II Channel (AWGN and two-path
Rayleigh Fading)
Fig. 4(a): BEP in n = 1 BMJ and AWGN
In this section, we present simulation results in BMJ and
Type II Channel (AWGN and two-path Rayleigh Fading).
Fig. 5 shows BEP of these three assignments corrupted by n
= 1 BMJ in Type II channel (AWGN and 2-path Rayleigh fading). It reveals that at the BEP level 10 −2 , “nvec = [2 0]” and
“nvec = random” assignments gain at least 15 dB over the
conventional FH/BFSK (“nvec = [0 1]”). The conventional
FH/BFSK performs much worse than the other two assignments in high Eb N J conditions, where the performances are
dominated by the multipath fading. The main reason why the
conventional FH/BFSK has poor BEPs is that the multipath
interference caused by the delayed second path destroys the
orthogonality between both FH/BFSK symbol tones. In the
conventional FH/BFSK, since both symbol tones are always
fixed contiguously in a common CD band in each hopping
interval, the resulting interference due to the non-orthogonality
severely degrades performances. For “nvec = [2 0]” and “nvec
= random” assignments, however, the chance that two
FH/BFSK symbols are contiguously located is greatly reduced.
Therefore, both these two types are more resistant to the multipath interference. On the other hand, Rayleigh fading is also
a contributing factor, which degrades all three assignments in
high Eb N J conditions. This factor is revealed by the comparison between Fig. 5 and Fig. 6. In Fig. 6, we remove the
effect of Rayleigh fading (i.e. both path amplitudes are set
with 1), but there is still a random time delay, IJ 2 ∈ [0, Tb ) ,
between the first path and the second path. Fig. 6 shows that
these three assignments have slightly improved performances
in high Eb N J conditions.
Fig. 4(b): BEP in n = 2 BMJ and AWGN
We further conduct another experiment where there is no
delay between the first path and the second path (this phenomenon, however, does not occur in realistic environment).
The corresponding results shown in Fig. 7 tell us that there are
almost no performance differences among these three assign-
Fig. 5: BEP in n = 1 BMJ, AWGN and 2-path Rayleigh fading
We here follow the analysis approach presented in Chapter
2, Part 2 of [1], where the jamming signals are assumed to
dominate over the AWGN. The conditional BEP is determined
only by p( m | nvec, qvec) and α . (i.e. the ratio of S to J/Q).
We assume p(BEP | m) = 0 for m = 0. By equations (2.36)
and (2.44) of [1], the simplified form of p(BEP | m) for m = 1
is (1/ 2) ⋅ u−1 (1 − α ) , where u−1 (•) denotes the unit step function. For m > 1, p(BEP | m) is simplified to be:
p(BEP | m) = (1 − m / 2) ⋅ u−1 (1 − α ) + m /(2π ) ⋅ cos −1 ( α / 2) .(7)
Fig. 6: BEP in n = 1 BMJ, AWGN and 2-path, no fading (equal gain)
In n = 2 BMJ, α ≥ 2.52 for Es N J > 1 . So (7) is simplified
as C ⋅ p(2 | nvec,[0 Q / 2]) , where C is a positive constant.
This indicates minimizing the BEP in (7) is equivalent to
minimizing p (2 | nvec,[0 Q / 2]) . Since
p(2 | [ 2 0] ,[0 Q / 2]) = {Q (Q − 2)} { Nt ( Nt − 2)} ,
(8)
p (2 | [ 0 1] ,[0 Q / 2]) = Q Nt ,
(9)
p (2 | random,[0 Q / 2]) = {Q(Q − 1)} { Nt ( Nt − 1)} ,
(10)
it follows that (8) < (10) < (9). When SJR gets high, (8) and
(10) decrease much faster than (9), which means the asymptotic performance curves of “nvec = [2 0]” and “nvec = random” go down and diverge from “nvec = [0 1]” in high SJR
conditions.
REFERENCES
Fig. 7: BEP in n = 1 BMJ, AWGN and 2-path Rayleigh fading, no delay on the
second path
IV. CONCLUSION
In this paper, we investigate a generalized signaling method
for non-coherent orthogonal FH/BFSK in BMJ and two types
of channel: Type I channel: AWGN only, and Type II channel:
AWGN and two-path Rayleigh fading. In the generalized
FH/BFSK, we let both symbol tones not be restricted within
one common band, but be pre-designed or randomly selected
from all available tones. In FH/BFSK in n = 2 BMJ and Type I
channel, the proposed two assignments gain around 6 dB over
the conventional FH/BFSK when the BEP level is 10 −2 . At the
same BEP level, they also gain at least 15 dB in n = 1 BMJ
and Type II channel. These numerical results show that both
symbol tones being located in different CD bands results in
much more resistance to jamming and multipath fading. The
complexity of generalized FH/BFSK grows linearly with the
symbol alphabet size, and this design can be easily integrated
by digital implementation.
APPENDIX
Asymptotic Behavior of p ( m | nvec, qvec ) in n = 2 BMJ
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