Transcript
Calhoun: The NPS Institutional Archive Theses and Dissertations
Thesis Collection
1984
An analysis of jamming effects on non-coherent digital receivers. Joo, Hae-Yeon Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/19138
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DUDLEY KbOX LIBFARY NAVAL MONTE;. [3
NAVAL POSTGRADUATE SCHOOL Monterey, California
THESIS AM ANALYSIS OF JAMMING EFFECTS ON NONCOHERENT DIGITAL RECEIVERS
by
Hae Yeon Joo December 1984
Thesis Advisor:
Daniel Bukofzer
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An Analysis of Jamming Effects
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PERIOD COVEREO
Thesis
December 1984
on Noncoherent Digital Receivers AUTHORS.)
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Naval Postgraduate School Monterey, California 93943 II.
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Jamming Effects on Noncoherent Receivers
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ABSTRACT
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The effects of various jamming waveforms on conventional binary incoherent digital receivers was analyzed in terms of resulting receiver performance (i.e., receiver probability of error Probability density functions associated with the test statistic generated by incoherent receivers under the influence of noise and jamming have obtained. )
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Due to the complexity of the mathematical expressions specifying receiver probability of error in closed form, no attempt has been made to obtain absolute optimum jamming waveforms operating against binary incoherent receiver. Therefore near optimum jammer signals were proposed, studied, and evaluated. The effect of a varying threshold on receiver performance was investigated and a jamming strategy involving use of an FM jammer was considered, and its effect evaluated. Graphical results are presented that highlight the mathematical results obtained.
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An Analysis Of Jamming Effects on Noncoherent Digital Receivers by Joo, Hae-Yeon Lieutenant^ Republic of Korea Navy B.S., Republic of Korea Naval Academy, 1979
Submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING from the
NAVAL POSTGRADUATE SCHOOL December 1984
.
DUDLEY KNOX LIBRARY P
'
MO-
ABSTRACT jamming waveforms on conventional
The effect of various
binary incoherent digital receivers was analyzed in terms of
resulting receiver performance
(i.e.,
receiver probability
of error)
functions associated with
Probability density
the test
statistic generated by incoherent receivers under the influence of noise and jamming have been obtained.
Due to
the complexity
mathematical expressions
of the
specifying receiver probability of error in closed form, attempt has
been made
waveforms
operating
Therefore
near
to obtain
absolute optimum
against binary
optimum
jammer
incoherent were
signals
no
jamming
receivers.
proposed,
studied, and evaluated.
The effect of
a
ance was investigated
varying threshold on
receiver perform-
and a jamming strategy
of an FM jammer was considered,
involving use
and its effect evaluated.
Graphical results are presented that highlight the mathematical results obtained.
-,
TABLE OF CONTENTS
I.
INTRODUCTION A.
COHERENT CORRELATION RECEIVER
9
B.
JAMMING OF COHERENT RECEIVERS
11
C.
II.
III.
IV.
9
1.
General
11
2.
Jammer Optimization
12
3.
Optimum Jammer Waveforms for PSK, FSK and ASK
15
INCOHERENT RECEIVERS 1.
General
2.
ASK (On
3.
FSK
17 -
Off keying)
18
19
ANALYSIS OF JAMMING ON ASK
21
A.
GENERAL
21
B.
ANALYSIS WITH NEAR OPTIMUM JAMMER
21
C.
VARIABLE THRESHOLDING EFFECT
30
ANALYSIS OF JAMMING ON FSK
32
A.
GENERAL
32
B.
ANALYSIS WITH NEAR OPTIMUM JAMMER
33
C.
FREQUENCY MODULATION SWEEP JAMMING
44
ANALYSIS OF FSK IN THE PRESENCE OF JAMMING AND FADING
V.
17
51
A.
GENERAL
51
B.
ANALYSIS WITH NEAR OPTIMUM JAMMER
51
DESCRIPTION OF GRAPHICAL RESULTS A.
GENERAL
B.
ASK (ON
55 55
-
OFF KEYING)
55
VI.
C.
FSK WITH TONE JAMMER
56
D.
FSK WITH FM JAMMER
58
CONCLUSIONS
APPENDIX
A:
60
DIGITAL COMPUTER IMPLEMENTATION OF THE
MARCUM Q- FUNCTION
72
LIST OF REFERENCES
76
INITIAL DISTRIBUTION LIST
77
LIST OF FIGURES
6.1 6.2
Correlation Receiver for Binary Signals Quadrature Receiver for ASK
6.4
Alternate Form of Quadrature Receiver Incoherent Matched Filter Receiver
6.5
Incoherent Frequency Shift Keying (FSK)
6.3
61
.62 62 63
Receiver
63
6.6
Modified Incoherent FSK Receiver
64
6.7
Near Optimum Jammer for ASK Modulation
65
6.8
Variable Threshold Effect on ASK Modulation
6.9
Near Optimum Jammer for FSK Modulation
67
6.10
Single Channel Jamming for FSK Modulation
68
6.11
Variable Threshold Effect on Single Channel
....
Jamming for FSK 6
.
12
=
1)
70
Frequency Modulated Jammer for FSK Modulation (K
A. 1
69
Frequency Modulated Jammer for FSK Modulation (K
6.13
66
=
2)
Rician Density Function
71 73
ACKNOWLEDGMENTS I
wish to
gratefully
Professor Daniel Bukofzer,
acknowledge
my thesis
advisor,
who provided help and assistance
in the completion of this thesis. I
also would
Stephen Jauregui
like to express my ,
Jr.
gratitude to Professor
for his support.
INTRODUCT ION
I.
A.
COHERENT CORRELATION RECEIVER The optimum receiver which will detect the presence of a
signal or discriminate between two
different signals in the
presence of additive white Gaussian noise, is the well-known coherent correlator receiver structure which has two equivashown in Figure 6.1 [Ref. 2]. binary communication problem is
lent forms,
The
as
hypothesis testing theoretic principles in signals,
or
(t)
S,
(t),
is
using
which one of two
received in the time interval
the presence of the
Due to
[0,T].
S
modeled
noise,
the observable
signal r(t) takes on one of the two following forms
H,: r(t)
S,(t)
=
+
O^t ^T
n(t)
or HB
:
r(t)
=
S
(t)
+
n(t)
0£t
-0)E/U
y0 3d
"
,
d
>0
,
d
=
,
d
<0
=
<0
and
9Pe 2 3d
-
J±Jk
S^ (d
+
/2tT
for all values of d. it
-
exp
is apparent
Since Pe
that making d
«$h a
is.
as
]
>
o
monotonic function of d,
large as possible in magni-
tude results in the largest possible increase in Pe
limit as
|d|-->co
,
is
it
However, from the Cauchy
-
easily seen
Schwarz
13
that Pe
inequality
.
In the --> 1/2.
)
(1.4) (n ;> s
l
with equality if n.(t) = /|2 where |n. |
I
,
|
j |
d
|
—>
»
'
is
J
proportional to
S^
(
Defining
t
jammer ener gy
the
is
P^.
the term
j
I;
when
< co
Since
possible to have infinite jammer energy,
one must
implies that P n
oo
S
> oo
- -
.
|
|
Sj
|
|
•J
it
is not
constrain
n.(t)
such
that
Cauchy-Schwarz inequality,
From
finite.
is
P,,.
the
Equation 1.4 can be made into an
equality if n.(t)
k.S d (t)
=
where K is a constant of proportionality.
If
|
|n.| -J
I
K must be set to the value 7
P^.
/
1 |
Sj
|
)
.
Thus
|d|
/P M j is maxim|
=
.
,
ized by setting
(1.5) J
/^Ts
and from
the above
n Ct)
=
d
(t)/
||S,
discussion,
results in
this
maximized also. Substituting Equation 1.5 into ability of error expression (Equation 1.3) yields ISrf
Pe
=
Pe being
the prob-
(1.6)
dx -co
=
-Jsnr(Ji-P t v^jsr)
CO
r 7
/27C
e
- x*/2
(1.7)
-x*/2
L.
<2
dx +//2^T
dx
Analysis of Equation 1.7 indicates the fact that for a value
beyond
> so-called 'break point', that is JSR (l~P)/2, an increasing SNR causes Pe to become worse, i.e., it increases
of JSR
3.
O ptimum
a
Jammer Wavef orm s for P SK
,
FSK and ASK
deterministic jammer waveform on various modulation techniques will now be presented for PSK modulation, where The
effectiveness of
S,(t)
=
A sinW c t
;
a
(t)
S
=
with the constraint that WcT
A sin(W c t n
=
j\_
K)
+
O^t^T
n an integer
\
results for FSK modulation will be shown, where
S,(t)
=
A sin W,t
;
S
(t)
=
A sin W
15
t
O^t ±T
Also
,
with the constraint that m7L
1
*,
S
;
(t)
where we assume that A
-
W )T
=
1 71
(W,
\
+
W )T
=
and m integers. Finally for ASK modulation
S,(t)=A,S(t)
>
(W,
|S|
|
<
|
co
and for convenience, that A
be obtained using
performances can
(
waveforms and resulting receiver
The optimum jammer
.
O^t^T
A S(t)
=
Equations 1.5
and 1.7.
Thus the optimum jammer for PSK is given by
2
nAt)-JPn Since
p
cos W c t
l-^r
-1 for PSK,
=
^t £ T
we have r -/Sir
(i +
J~^)
z
P
=
- x /2
—
'TlL
dx +
dx
J -co
2SNR (1 V-JSR)
For FSK modulation, the optimum jammer is given n (t) j
=
/p
tij
J= V
-WJtcosi
aan^CW,
(W,
+
W„
)t
1
Using the assumptions on W Thus Equation 1.7 becomes
and W
the value of
,
-,/§NR ^«R (1 + 1
2
- x "/?
O
7C
y^
^jsr
is
zero.
/2JSR
)
-x*/2 dx
(i -
/0
(
+
/y2
7U
fc-
dx
-°° )
Finally, for ASK modulation, the optimum jammer is given by S(t)
n.(t)
=
/P,
and the probability of error becomes
16
s
f
Jim
-
TC
dx
(v^ a
-x
V2JSR)
+
/2
dx
-co
S"R (/^-/2JSR)
where now 2A,A
P-
At
+
At
and (A,
a-The
=
Af
'break point'
-
A
+
Ao
fl
)
for ASK occurs at JSR
'break point'
in terms of
1,
a
-p
i
occuring at JSR C.
with
in efficiency
lowest
is
^
<
FSK is next highest
1.
occuring at
in efficiency with a 'break point'
ASK
Since
0^/2.
efficiency, PSK is highest with
'break point' occuring at a JSR of
and
=
a
JSR of 1/2, 'break point' a
1/2.
<
INCOHERENT RECEIVERS 1.
General In the coherent systems considered,
bearing
signals were
assumed to
In noncoherent
systems,
be known
the information
exactly at
the
receiver available at the receiver so
signals is not is
treated as
[0,27C].
incoherent
As
phase of
the
a
random variable uniformly
such,
system
we to
may expect be degraded
the carrier
that the phase
distributed over
the performance in
comparison
of an to
the
performance of the corresponding coherent system. However, because of their simplicity, incoherent systems are widely used in many applications.
17
ASK (On
2.
Off keying)
-
incoherent ASK (On
case of
In the
-
keying),
Off
the
receiver in the presence of
signals at the front-end of the
noise only can be mathematicaly modeled by either
H,
:
r(t)
=
A sin (W
Q)
(1-8)
/ / O^t^T
n (t)
+
or H
O^t ^T
the frequency Wc and the
The amplitude A,
are assumed to be known except
time of arrival
that the phase
(B
modeled
is
as a random variable having an a priori density function
0^6
27c %>
(8)
=1
£27^
0.
otherwise
The additive noise is again assumed
to be zero mean,
Gaussian, with power spectral density level N
/
2
white
watts/Hz.
waveform n.(t), the optimum receiver for decoding the binary information (which is transmitted via the use of either a sinusoid that is incoherently received, or no signal at all) in each interval absence of
In the
of duration T
sec,
jamming
a
the well-known
is
Its structure is shown in Figure 6.2
There
is
an
alternate
receiver obtained by replacing
quadrature receiver.
[Ref.
form
of
2].
the
the correlators with matched
filters having an impulse response given by h(t) -
t)
and h(t)
outputs at
t
cos Wc(T
=
=
T
-
t),
(Figure 6.3)
quadrature
<
[Ref.
t
<
T,
18
sin Wc(T
and sampling the
2].
Still another important alternate
rature receiver is an incoherent
=
form of the quad-
matched filter followed by
an envelope [Ref.
detector and
sampler
a
Figure 6.4
shown in
as
2].
FSK
3.
binary
For
received signals
keying
shift
only can
the
(FSK),
the receiver
front end of
at the
of noise
presence
frequency
in the
mathematically modeled
be
by
either H,
:
r (t)
=
A sin (W,t
r(t)
=
A sin (W
+0)
+
n(t)
+
n(t)
O^t^T
or H
:
with
Gaussian,
)
^t
power
spectral
Usually the frequencies W
watts/Hz.
^T
noise is again assumed to
where the additive white
t + Cj)
be zero mean,
density level
N /2
differ substan-
and W (
The phases
tially.
random variables,
and
are statistically independent
(h
uniformly distributed over
the interval
[0,27H].
amplitudes is concerned,
Insofar as the signal cases are distinguished.
bearing
treated as
are known
the information
second case,
the
and
equal.
independent random
variables that propagation.
For this second
used :
r(t)
=
A sin (W,t
H
:
r(t)
=
B sin (W t +
+
the
are Rayleigh
following hypothesis testing theoretic
H,
In
bearing signal amplitudes are
distributed due to multipath case,
the information
In the first case,
signal amplitudes
two
0)
n(t)
+
^t ^T
and
)
19
+
n(t)
O^t^T
model is
.
we assume
where
that
remain
random,
the
signal amplitudes
constant over
a
.
T
-
second
even
though
interval.
The
noise is modeled as in the previous analysis, and the signal
amplitudes have
(a
priori)
probability density
functions
given by a
exp (-
=
fA (a)
-J5j
2.
75)
3.
>
and b f (b) B
where A 2
=^iexp(=
,
,*
E{A 2
=
E{B 2
results
The
receivers
}
can
be
}
associated
applied
incoherent FSK receiver. The resulting
toward
be essentially
ASK
obtaining
incoherent the
shown in Figure
filters shown in Figure
replaced with tuned bandpass
center frequencies t
with
optimum
This is worked out in Reference
receiver structure is
Individual channel matched
frequencies W
b>0
-fjz)
20
.
6.5 can
filters having
corresponding to the "mark"
and Wo.
6.5
2.
and "space"
ANALYSIS OF JAMMING ON ASK
II.
A.
GENERAL the effect of a
In this chapter,
waveform on
the performance of
deterministic jammer
an incoherent
ASK receiver
will be investigated. The signal at the front-end of
Equation
with
1.8
hypothesis,
a
the receiver is given by
modification
the
jamming waveform n (t)
that
under
either
is present during the
;
time interval [0,T].
For
constrained
Equation 1.5
ASK receiver,
coherent
the
jammer .
waveform
the
optimum
obtained
could be
by
energy
using
However, the complexity of the mathematical
expression for probability of error in the noncoherent case makes difficult if not impossible to derive the it very optimum energy constrained jammer waveform in closed form as will be seen in the sequel.
reasonable postulation
A
is
that
the optimum
jamming
waveform for the coherent ASK receiver can act as a good jammer for the incoherent ASK receiver also. Thus, such near optimum jammer signals are studied and evaluated in the performance of the noncoherent
terms of their effect on
ASK receiver.
B.
ANALYSIS WITH NEAR OPTIMUM JAMMER Analysis
of the
incoherent
mathematical model of the
receiver
starts from
receiver front-end input waveform
r(t) given by either H,
:
r(t)
=
A sin (Wc t
+
the
Q
)
+
n(t) + n.(t) j
or
21
£t IT
.
H
r(t)
:
n(t)
=
+
O^t^T
n.(t) j
where n.(t)
the jammer
is
waveform present during the time
interval [0,T] In the absence
problem is well-known.
binary ASK
the statistical
documented in [Ref.
the optimum
of n.(t),
receiver for the
Its derivation
detection theory
is well
literature
The receiver structure is shown in Figure 6.2
2].
In this section,
the effect of n.(t) on this receiver is
analyzed by evaluating the resulting
Pe
,
under the assump-
tion that n.(t) is a deterministic jammer waveform,
however
sJ
unknown
receiver
the
to
Receiver
itself.
requires determination of the statistics
where G
2
is
of either G 2
quadrature detector
output of the
the
performance or G,
and is
given by X
1
G
=
X
3-
Y
+
where
X
= Jc
Y
r(t) sin W c t dt
T
r(t) cos W c t dt
=
Z
(r,S)
=
(r,C)
(
Provided that the random variable 9
,
X and Y are conditional
E{X|H, ,8]
=
£p
A
m >|9
and ,C)
fixed to some value
Gaussian random variables with
(AS a ,S) + (n.,S)
{y|h,,0}= (AS e
is
+ (n ;
,C)
22
I
m
y(e
where
used in place of
is
Se
sin(Wc
+
t
Q
It
).
can also
be seen that a
Var {x|H
E {(n,S)
= (
}
,QJ
sin 2W C T
N.T
2Wt_T
=
Var {x|H a J
and similarly ~l
sin 2ty.T
= Var
>WC
T
=
n 7C
{y|h }
we assume WcT
For convenience
Thus the sine terms above =
var{Y
t-{,
,
^
=
}
N T/4
|
=
|-|.
,
2
.
term is
additional sine
where n is an integer.
vanish resulting in var{X } If this assumption is not (J
an additional term results.
made, the
|
,
However,
if
may be
small and
7C
/Wc << T,
neglected.
Furthermore the covariance 1 X - E {>'IH.,0}
{y|h.,0
Y
•
H;.0
E |(n,S) (n,C)j N T
1 -
cos 2W C T
4
=
i = 0,1
2W t T
_
provided the assumptions on Wc any
given value
of
,
both
Gaussian random variables and pendent.
phase
The is
This implies that for
hold. X
and
Y are
uncorrelated
therefore statistically inde-
density function
of G
2
conditioned on
the
noncentral Chi-Squared distributed, and is given
by
23
)
^c«|H„e,.^.^-ii£!^E) The variance
otherwise.
and zero
2
(J
,
g>0
is defined
above and
becomes (2.1) S = m*ie
[(AS»,S)
2
Z
m*„
+
+
{x|H,,9}
E
=
S
f AT
=
\
\^2~) +
(n
=
r
AT [(n.,S) cos
+
,sf
+
(n
{y|H,,g}
f(AS e ,C) + (n^C)"]*
(nj,S)f+
Due to our assumption WcT
+ E
r\7t
9
,
we obtain
-i
+
(n. ,C)
sin
,cf
Similarly, if r(t) consists of noise and jammer only, then X and Y are also independent Gaussian random variables with E
{
X|H
|
(n.,S)
=
4
m.
=
and
e|y|H
= |
(n.,C)
=
mY
The density function of G 2 assuming no signal is sent is
V
(,|H.,.
_L ^ {
.^
)l
M
and zero otherwise, where S
'~< +m r
=
Cn., S f
+
(
n .,cf
24
(2.2)
The resulting average probability of error is given by
P*
pp-P(HJ+ PM P(H.)
=
P(Ho)
V
/
( 8l
Ho
x
dg
)
|fy
+ P(H.)
(
g H| |
)
di
-00
is the threshold with
which G is compared in order to decide which is the true hypothesis. Observe that
where
Tj
fjzCglH.)
f^ |e
=
of false
(probability
f
9
(0) dQ
the above expression
integral in
The first
0)
(g|H,,
alarm)
It
.
can be
for Pe
is
expressed
Br
as
follows oo
g + 2o
x
exp C-
s o
2
2
-)
I
(
^=-
dg
)
n exp (-
V
liil)
i
(
a 'v)
dV
/7 ?/(J
Q
a',
(
n/o
)
where .CD
Q
a
(
,
6
)
V
=
exp (-
Q
V t 2
# is
a
well-known
Q- function.
Pe is
PM
function
tabulated
called
the
Marcum
The second integral in the above expression for
(probability
of miss).
It
can be
expressed as
follows tv-~ r *
V
H, ,8) G *| Q (g|
f g
(8)
25
d9
dg
(2.3)
r
rco
r^J-
^f
(8)
f
(g|H, ,6) G
the second equality
where in
de
J, *
L-
-co
dg
8
|
the integration
the order of
The inner integral of Equation 2.3 can be
has been changed.
expressed in term of the Marcum Q- function as
"^i
£_l_i)
exp (-
a;
2
(% =
-
1
where q( = JS//T written as
V
exp (-
Q
a
(
(€L)
I
dg
a
2
+ a
n / a
,
)
l fl
aV) dV
(
)
Therefore the probability of error can be
.
7
P
P(H
=
e
)
/s (—
Q
',
(2.4)
n/o
)
f2 7T *
+
P(H.)
1
2
y^ (~
Q
n /o
,
di
)
TT
so
dependence on Q is imbedded in the term / S and for the jammer absent case the threshold Tj is obtained as the solution to the equation
where the
-
A T/2N
Q
I
(,2An/N
=
)
R
R
5
~
PCH
)
P(H,
)
which can be equivalently put in the form -
A T/2M
e The
I
( • )
conjunction
hA
T/N-
(
n
/
function used here
with the
a
)
R
and also previously
development
26
=
of
Pe is
the
used in
modified
Bessel function of the first kind. of the threshold is an important
well as noncoherent receivers.
The appropriate setting issue for both coherent as
could use the threshold
One
setting that would be derived from the analysis of receivers
operating only.
strated
additive white
in
This approach can be in Reference
obtain Pe as
a
Gaussian
interference
quite unsatisfactory as demon-
would be
better approach
A
1.
noise
function of the threshold,
to
and then search
for the threshold that minimizes Pe.
While this approach is
intuitively appealing,
mathematically intrac-
table.
is often
it
threshold issue is not
While the
addressed in this
discussed in more detail in the sequel and simulation results will be presented. definition of average signal energy previously If the
particular section,
it will be
introduced is used,
we have E
A 2 T/4- which
=
transmission case, bearing signal(s) do
to the binary signal
half in comparison
fact that the information
due to the
is reduced in
not have equal energy. the coherent receiver
In order to afford comparisons with we will implicitely
case,
value of signal amplitude A, to obtain
E
=
boost the
A T/2 in order to 2
have agreement with
previous cases insofar as signal energies is concerned. Thus the threshold determination equation
now becomes -SNR
e
(2.5)
P(H„) [\/2SNR
(
If we assume P(H.)
-
L
»J
n
=
/a
)]
P(H.)
=
=
R
R
=
F(H,
then
1/2,
iri
u
and the threshold setting equation becomes
-SNR
Q
f
I
[/2SNF (1 /„
)
=
)J
27
l
/a
)
If
we are
jammer waveform
the optimum
to find
to
so as
maximize Pe, an attempt must be made to solve
—
=
and
Unfortunately
as
the
,
=
resultant equations
readily solvable for n.(t).
involved and do not appear seems however
mathematically
are
that a 'good'
jammer waveform can
It
be postu-
lated based on the results obtained for coherent ASK. It was
found for that case that the optimum n.(t)
a tone at
is
the
o
carrier frequency. Thus the following jammer waveform can be used as a potential near optimum jammer, namely
.
/
p
Observe that with ability
of
error
threshold setting
f
this choice,
||n.||
2
P*.
=
J
Pe
can
now
equation (Eqn.
derived expressions for
S
and S'.
.
2
6
")
The prob-
J
determined
be
and
2.5)
using
the
the previously
The effect
of the near
waveform on the receiver (i.e., incoherent receiver performance) can be analyzed by evaluating Pe as a function of n.(t) using Equation 2.6 Note that in Equation 2.6 the jammer energy is P^. It can be shown that
optimum jammer
.
.
(n.,C) =f
r
/n~/X
sin W r t cos VLt dt
=
Then from Equations 2.1 and 2.2, the probability of error Pe given by Equation 2.4 can be expressed in terms of SNR (signal-to-noise ratio) and JSR (j ammer- to- signal ratio)
using the fact that
28
S
~qJ= "tS;
=
/
o 2
SNR (1 +
2
SNR- JSR
and the fact that
Tj
equally
For
2
/JSR cos9+ JSR)
be obtained from Equation 2.5.
can
/^-
likely hypotheses,
the
probability
of
receiver error (Equation 2.4) can be expressed by (2.7) i-Fl
/2SNRJSR
(
^/(J ) 2 v/JSl?cos9+ JSR ,
+J3
__L.(^(
/2SMR (1
+
,"7/^-
(a ,3
Q
(2.8) Q
(a',6
di
)
where
/2SNR
#
^2 SNR JSR
Since the
=
cause the
pi
=
J
'
2
,
=
7?
/
1
,
x
Equation 2.12 can be rewritten as (2.13) 4
yj
=
a' -
2SIIR 2
From Equation
2
+
j2f»2TT
2
.
13
,
j^a'
J?rK2SNR)
it
+
USMR
+
fa
R
be recognized that when SNR is
can
variation of the value of R does not significantly affect the threshold value qC because taking the logarithm of R reduces the effect of R further. Thus the large value large,
'
variation of the term J2n R on ^' This limited variable thresholding effect on the receiver which is under significant jamming is analyzed in graphical form in Chapter 5 for various values of R.
of SNR suppresses the effect of .
31
III. ANALYS IS OF JAMMING ON FSK
A.
GENERAL incoherent FSK with
For binary
a
jammer
present,
the
received signals under the two hypotheses are either H,
:
r(t)
A sin (W,t
=
+
*t ^T ~
n(t) + n.(t)
+
)
j
or W
:
r(t)
=
A sin (W
t +
By separating the frequencies W,
form signals that
It iT
n(t) + n.(t)
+
(f))
sufficiently, we can
and W
have equal
are orthogonal,
energy,
and
have the same advantage of ease of generation. The modified FSK receiver structure
varying the output followed by
a
which is capable of of the each envelope detector (which is
multiplier) is diagramed in Figure 6.6
.
The
optimum receiver for the case where no jammer is present can be derived from statistical decision
Figure 6.5
.
theory and is shown in
This receiver can be obtained by setting
1/2 in the modified FSK receiver shown in Figure 6.6
FSK is widely used
incoherent
In practice,
its simple receiver structure,
c(
=
.
because of
its small performance penalty
due to lack of phase coherence and its more efficient use of
signal energies in comparison to tion,
we
are not
faced with the
threshold that
must change
incoherent ASK.
Thus,
a
incoherent ASK.
In addi-
difficulty imposed
with SNR
the case
as is
receiver which
is
known
by a
with to be
optimum for incoherent FSK transmission, has been modified by including some channel weighting. This has been done in order to be able to determine whether or not such channel
weighting can reduce the effect of the jammer. 32
This chapter is devoted to investigating the performance of the modified
incoherent FSK receiver in
the presence of
jamming and additive white Gaussian noise. can as
1/2 we
a
byproduct
conventional incoherent
obtain the
By letting
performance of
FSK receiver of
Figure 6.5
=
pC
the
in the
presence of jamming and additive white Gaussian noise. ANALYSIS WITH NEAR OPTIMUM JAMMER
B.
modified receiver
The
introducing
a
hypothesis and analysis
of
performance can
hypothesis (no
null
by following
incoherent
signal)
the same
as
the
in
by
dummy
a
reasoning as
presented
ASK
be obtained
in the
previous
chapter. The receiver function is to compare the envelopes at the T seconds
each channel once every
output of
decide in
and
favor of the larger of the two envelopes (Figure 6.6). the purpose of analysis,
let us
signal has been transmitted,
assumed to be true. An error is is
the hypothesis H, is
An error is committed if V is
V|
exceeds
larger than Wq
assumed to be true [Ref.
Let
Pe,
Under the
sent,
the probability
denote
first type
of the
the assumption
output q
that a
of one of
'mark'
signal
the envelope
where
=
r(t) sin W,t dt
;
Y, =
33
/
>
of
V
(
|
has been
detectors is
f
given by
X,
when a
3].
error described above, which is expressed as Prob.(V H.).
V|
signal has been transmitted, that is, the hypothesis
'space'
H
first assume that a 'mark'
that is,
also committed if
For
r(t) cos W,t dt
Observe that X
and
(
conditioned on the
Y,
phase and either
of the two hypotheses are Gaussian random variables with
T E |x,|H,,0}= T
+
f
/ A
sin (W,t
+
n.(t) sin W,t dt
sin W,t dt
)
(S,,S),
=
(n.,S),
+
and
E
H,,0}=
{'Y,|
+
where
I
/A sin (W,t
n.(t) cos W,t dt
=
cos W,t dt
)
+ (n.,C),
(S, ,C),
represents the function A sin (W
S|
sents the function sin W,t.
Q
+
W| t
(
is
6
),
S
repre-
n7L, n an integer,
=
(
n(t)
+
and C represents the function cos
Likewise, assuming again that W T
and that
t
white Gaussian noise
zero mean
with PSD
level N^/2 watts/Hz, we obtain Var [x,|H,,eJ
=
Var
[yJh^G}
^
=
Furthermore, it can be shown that e
[x,
- e {x,|h,
,e}][y,
-
|
the conditional r.v.'s
so that
and therefore independent.
ables X, and Y,,
eJy.Ih.^Ih^oJ and
X,
Y|
=
are uncorrelated
sum involving random vari-
The
and producing q*,
will result
in a non-
central Chi-Squared distribution so that
f
0>
fq,|H,,6)
^lexp(-
=
^)I„<^> 2
o
where S„
=
E^XjH,,©] (S, ,S),
+
+
E*{y |H,,0] 1
(n.,S),[
+
34
[(S,
,C),
+
Cn.,C),]
q,
>o
and
2
N T/4.
-
(j
Using standard random variable transforma-
tion techniques, it is not difficult to show that
Wi,iH.,e> =^«xp(-lL!!)i o
fij^L)
q>2i0
so that
fO,
(cl.
H
I
>
-
W
=
-/a
(3.1)
% O)
(qjH.,6)
d0
o
where the dependence on 6 is imbedded in the term S M now need to obtain the statistics of the output of
We
.
fv
(^i
**
detector.
(See
density function
From standard transforma-
needs to be derived.
)
I
the probability
That is,
Figure 6.6).
envelope
the upper
following
multiplier
the
I
tion theory, using the relation
V,
=
2(1
-
(/.
)q
,
it can be
(
shown that f
(v,|H,)
=
2(1-
a)
47t(1 . a
fQl (
—
°
the other
detector when H
|
is
-} 2
a
de
'
2-
hand,
(3.2)
a
'{ iii^V 5 On
)
)
jhill^1
)
a-
|H,
2(1- a
the
)
output of
assumed to be the
given by
35
the lower
envelope
true hypothesis,
is
,
q„
=
X
Y
+
where r(t) sin W t dt
X,
Y-
;
r(t) cos W„t dt
=
similiar procedure
Following a
as used
above,
can be
it
shown that
|y
|H (
,
e|
/A sin
=
+
(W, t
n .(t)
I
6)
+
cos W
t
cos W D t dt dt
(S
=
,
,C)
(n. ,C)
+
Jo
and
e|Xo)H, ,ej
A sin
=
(W, t
+
sin W
6)
dt
t
Jo +
Here,
S
and
except that
nj(t) sin W
.
It
dt
=
(S, ,S)
+
(n. ,S),
the same meaning previously defined "0" outside the subscript the inner products C
have
implies that we should Wct
t
interpret
as sin W
S
t
and
C as
cos
can also be demonstrated that
Var
(
X e |H,,e}= Var
N T
|y o |H,,G}
and also that
e{
[
so that the
hence
X
-
E
[xjH.,0}]
Y
conditional r.v.'s X
independent.
expression becomes
[
for
the
Thus
-
E
D
similiar
conditional
36
{Y |H„eJ]|H,,e}=
and
Y
are uncorrelated
Equation 3.1, the density function of q to
>
f
(qJH.
Q
z
a
I
°»
s
<
2
exp (-
V Thus the average error probability is is made if V i
.
averaging the conditional error
found by
probability given
by f
=
P(V.)
(V |H,,V,) dV
A/.
over all
Vi
r
e,
.
That is
Prob,
v
,
>v, H,]
(3.4)
I
37
-
JTf
-
Jo L
f„
H,
(V,
dV,
)
(V,|H,
fu
V,
)
dV,
v,
Substituting for the density functions in Equation 3.4, from Equations 3.2 and 3.3, we obtain roo
271,
r<°
oa
7ZJ
2
(
-
^2a
dV
de]
f
(_2aVf^)
the order
|
b«
(3.5) \
'
(
)
dV,
been changed
for
In the above equation the inner-
expressed
can be
integral
(V,|H V(
integration has
of the
computational convenience. most
+ /
[_J
I
where
exp
terms of
in
Marcum
the
Q-function as follows
—
^~
exp (-
;
So
*
'
I.(
bo 2a V )^V a '
2a
V,
=
q
^K
<
a
V, )
/aa
'
Then, Equation 3.5 for Pe
rt
ZTT
o
L
-^0
*q (
L 2
(V,|H,)
f
2aa
dV,
(3.6)
V(
^Z
,
il£L x
)
di
/
IT
^7C I
L 2tt
'
-'O
3
J
becomes V.
vS«i
1 ei
)
x
exp
(
«* Of},
J
38
! .(
x
S
de
dx
where in
equality the
the second
V
variable x
change of
has been made
,
2(l-a)a
From the orthogonality
property of the signal
pair used
(which is obtained by assuming sufficient separation between two frequencies
and that Wl as
well as W
we
large),
are
have
f sin
(W.t +
)
sin W„t dt
term
that the
so
9
S
|
=
independent
is
of Q
Therefore
.
Equation 3.6 can be rewritten in the following form
P
x exp
= 2
e,
Ja
(-
+
I
—— 2
) \ >
•
L Jo
v 3,i
I. (x
where
X
t
)
order
the
dx
d9
integration
of
been
has
changed.
using the following formula involving an inte-
Furthermore,
gral of a Marcum Q-function [Ref. A]
Jn
1
-
<£**?
Q
(
Jn— Vo*+
a;
2.
o? *
grp^' Va,^-o- 'Vo^?i
the inner integral in Equation 3.7 can be simplified in such a
way that Pe,
becomes (3.8)
p
=
(1- a)
(ITaf
i
+a
:
_ 2tt
Q
u
39
V(l- af+a
a ,
a
\/
H
|
takes the form
)
(3.9) '±n
a
-
1
n-l
(1- af+a*
Q
7¥J
(1- oO+a a
(1- a)
(1- a)
1
1-a 7+ a 1
v/5,,
(f
j
(,_^)> +i<
a (1-a L \/s^ / V(l-a 7+a*- ,"5-/(1-
(
S
O-oO'
/ \/
/
o
where only the term
<
,a
/)V v)d?
dependent on
is
,»
a
^
(
Pe
- P ,
)
€e
assuming that the two hypotheses are equally likely. special case q( the performance of the 1/2, optimum receiver in the presence of jamming is given by For the
v4»-i
/T"
2tt
JL
(
^
a
,/7
v
^'T,vf'^\
^
2tt
^
f q
(
J.
40
a
+
JiX 4-
f(S
,S),
+
(n
S)J
+
[CS
+
,C) (
4-
41
(n.,C)J
the unmodified
error of
probability of
Thus the
can be expressed in
(Equation 3.10)
receiver
terms of SNR
and JSR
only (defined by Equations 2.10 and 2.11) as follows
V
f
x
-n
^jH
Q
(
Q
(
/f,/f> -q
c/p^jd.
(3.12)
1
-
,\T7
2
/¥>}**
Q
where
q(ix -
SNR (2
oLo-
SNR
$m=
p.o
+
2
J 2 JSR cos
(2-2 7 =
2JSR cos
9
<|>
+JSP) +
JSR)
snr JSR
Receiver performance can now be evaluated as a function of SNR for fixed values of JSR. To provide further insight into the performance of the modified incoherent receiver, the effect of varying q/^ can be analyzed via computation of the probability of error Pe from Equations 3.8
and 3.9.
In terms of SNR
and JSR,
becomes =
=
•
PCH.) Pe
,
+
P(H
p(h.) c,[ i -
)
.gL-
?Qo
(3.13)
q
(/cz^r^r,)
dej
At7l\^°^
^P^V^O
d0]
]
d«p]
and if the second choice is used, we obtain
Pe
P(H,) C
=
x
"
[
P(H,) C 2 +
P(HJ
C
+
P(H
C,
e )
-±^ -
1
^Co
-^rl Q
{
fcfwcip,',
Jc*fi*
JCit,.
-j^jQ (Jc^]
)
threshold
oC
)
^9
dcp_
on receiver perform-
jammer waveform of Equation
ance for the
de
d6
JCifio',
gTfjQ (/^.^ ,/c777)
The effect of varying the
)
3
.
discussed
14 is
in Chapter 5.
C.
FREQUENCY MODULATION SWEEP JAMMING In
certain
situations,
rather than
frequency band
a
the need
to
discrete
set of
jam
certain frequencies a
Therefore an FM sweep jammer
using tone jammers may arise.
will be proposed, analyzed, and its effect on the incoherent FSK receiver
investigated as
a
function
of the
number of
times the jammer sweeps the signal band during the signaling
interval [0,T].
The mathematical model used for an FM sweep
jammer is n.(t)
=
\fi-*i*
J
^t
+
K^ fa cos W.t dt]
[
After earring out the integration we obtain
n.(t) j
=
%Ph
We t
sin i
+
sin W.t A £
44
+
0^t£T
p
where
=
and
K_ptf./Wj
phase angl e
is a deterministic
g
The instantaneous jammer frequency is
Wx
(t)
^W-
-
Ws
,
where
27Lk,
o W
+
r
cos w#t
J
J
instantaneous jammer frequency covers
so that the (W s
Ws
=
1
Assume that W 5 T = 2 7C 1 and W. T = £ Wj ) and k are integers. In order to determine +
.
receiver performance in the presence parameters
S; K
(i,k
J>
S) *
=
which are
0,1)
=
of such a jammer,
the
function of n.(t),
a
o
Before so doing, we evaluate
must be determined. (n
the band
JjX^J^-sin
(
W st
+
^sin W.t) sin W k t dt K
=
,
(3.16)
1
where since the fixed phase & represents a time delay only, Equation 3.16 can it is set to zero for computational ease. be rewritten as follows
(3.17)
T
COS
(n.,S) K
(Wc,
-
w K )t +
VLt P sin
j
-
By using
cos
f(Ws + W K )t
+
dt
p sin W.t_
the well-known Bessel function
coefficient expan-
sion for each cosine term in Equation 3.17, the following is
obtained
Si^^-^-f-rnalT (n.,s) K
Z.1
(u^-u/kj T
—
stJO^+wQ-hiw.It ^s + w j t n vu 45
+ niu. *J
(3.18)
Since the
FSK signal
4/t/T
+
W,
,
covers approximately
(assuming W
47C/T)
>
t
W
the band
(Wo it is reasonable to
),
set
=j(W,
Ws
+
w
(3.19)
)
so that the instantaneous jammer frequency band \i
Q W«
+
ws
-
QW
.
,
completely covers the signal band provided that
)
£W.
-
(W5
=
w
-
-£t£
ws
j
+ (5W.
=
4-7C
W
This means that =-!-(
p.
- w„)
w,
must be satisfied.
+
4£ FSK signaling we have assumed
Since for
that
where
-
W,
(
1
w
)
T
=
17C
(W,
and m are integers,
w D) T
+
and W*T
=
m7TL
27Ck,
it is apparent
that
(8=
-gfc-[i.
Note now that
+
1/2]
the integer k determines the
the jamming waveform
Thus from
interval.
number of times
will sweep the signal band
Equation 3.18 and Equation
in one bit
3.19,
have r
(n.,S),
srnf^Cu;.-u; )+tiu;.jT (
J
2"(3^ + which becomes
46
U)+ 7)U/.
J
we
pI T
Z^
(n.,S), Z.
(
(3
(3.20)
Tl-.-oO
Sin 7c(^K-Mn-fi/£)
and
^_ £
(n.,S)
Sin[T(^-^ )+WjT
)("
Jfl(
p
J
which becomes
Bui £j,(p
(n.,S).
g.
j
-« =
->
(3.21)
If
1
then Equation 3.20
is an even integer,
and Equation
3.21 at most contain two terms respectively. These terms can all SINC functions
if the argument of
exist only
Therefore Equation 3.20 and Equation
is zero.
3.21 can be simplified
to yield
(npS),
where n,
-~l^r-
=
(n.,S)
l/4k, n^
:
In view
=
-
(m
- J* a
+
<
1/2)
(3)]
/
2k and
a
(
j
where n
P
[Jt,,(
l/4k,
n^
-
of practical
(m
1/2)
-
/
2k.
communication system
constraints,
the integer m will typically be much larger than
and
1
represent
frequencies
the sum
respectively.
and difference In
47
of the
particular
we
1
because m
signaling see
that
JK
(^)<<
J-n
(
1
for
<<
^
Hence the
k.
terms
J^
(
B
)
and
can be neglected so that
6> )
(n.,S),
JW
Cn.,S)
=/M
(3.22) J-n, < (»
)
and
Jl|j(
p
(3.23)
)
The other parameters to be considered are (n.,C) K =
where
=
Kr A
I
/
Jl^.p=rsin (W 6 t W-
+
sin W.t) cos W K t dt
(3
Using again the Bessel function coef-
.
appropriate trigonometric identities,
ficient expansion and we obtain
(3.24) (n.,C)
-f
-f(2nK-Je/2)
and "
CO
(n. J
,
C)
2
sin '-g-(2nk
+
1/2)
(2nk
+
1/2)
= ti = -cp
sin*-^(2nk
+
m -1/2)
(2nk
+
m -1/2)
48
(3.25)
We note
that all
Equation
the terms in
3.25 take on the form of
3
.
and Equation
24
SINC function times
a
a
sine func-
We can readily show that for even values of m and 1,
tion.
Cn.,C),
=
Cn.,C)
(3.26) = fl
o
Using the mathematical forms given the performance
and 3.26,
by Equations 3.22,
receiver with
of the
3.23
FM sweep
jamming is obtained via the use of Equations 3.8 and 3.9 and evaluated from the expression (3.27) p4
=pch.) + pch.)
c.ti.^JQcy^ c2
+ p(Ho) Ci
[-^JTc/cT^ [-he
$
{
J
yzr^)de]
JZZ
)
d e]
J^~° ./gZT)
V
where
dr oL= (&,=
2 2
SNR [ 1 + 2/jSRj^ SNR
(3,. =
[
2
1 + 2 7jSR
SNR JSR
(
J^C
£
)
p
)
J^(fi)
49
cos9
+
coscp +
JSR j-
JSR
j^
(
p
(
^
}
]
j
C
4
and
Cg,
have been defined in Equation 3.13.
this equation SNR and JSR are
As before,
in
defined by Equations 2.10 and
2.11 respectively. The performance results of SNR,
JSR,
signal band.
to be presented are
a
function
and the number of times the jammer sweeps the This is discussed in greater detail in Chapter
5.
50
.
IV.
A.
ANAL YSIS OF FSK IN THE PRESENCE OF JAMMI NG AND FADING
GENERAL In certain propagation media, the received signal trans-
mitted via a
free space channel
a
is often subject
to fading,
phenomenon caused by multipath propagation and the equiva-
lent addition of random phasors.
envelope which
have an
The vector sum signal will
changes with
time resulting
in an
effect known as fading. signal model often
The fading
nonselective,
fading,
slow
amplitude where
it
is
utilized is that
of the
distributed
signal
the amplitude,
while
Rayleigh
assumed that
Thus
random, remains constant over the time interval [0,T].
the received envelope is now random with probability density
function f A (a)
The
E
E
where £ = E{A amplitude
a>
_±£-exp(- Jl.)
=
A
2
}
denotes the mean squared value of the signal
optimum (minimum
probability
structure in the
presence of additive white
(however with no
jamming present)
nonfading FSK
noncoherent
given by the receiver of Figure B.
Gaussian noise
the same
is
structure is
and its
receiver
of error)
as that
for
therefore
5.
ANALYSIS WITH NEAR OPTIMUM JAMMER We shall
incoherent
jammer in Thus,
the
analyze the FSK receiver
addition to signals at
effect of
operating
in
the additive
the front
either 51
Rayleigh fading the presence
white Gaussian
end of
on an of
a
noise.
the receiver
are
H,
r(t)
:
A sin
=
+0)
(W, t
^t ^T
n(t) + n.(t)
+
J
or
H
B sin
=
>(t)
(VJ
t +Cf>)
amplitudes A
where the
+ n(t)
O^t^T
n.(t)
+
are independent
and B
identically
distributed random variables having density functions
f
f
A
B
with E{A}
(a)
-^bexpC-
=
(b)
_b_
-fcexpC--^)
=
E{B}
=
a
2 A,
=/
b>:0
and E{A 2
-g A
Note that FSK with fading is ously analyzed
signal
>0
(FSK without
=
}
E(B 2
=
}
the same as what was previ-
fading)
except
random varibles.
amplitudes are
2A;
that now
the
Therefore for
a
fixed value of A, the error probability when a "mark" signal has been transmitted is the same
(nonfading)
FSK case.
as that of the noncoherent
we can use the results of
That is,
the preceding chapter and in particular make use of Equation 3.7,
to obtain (for the special case of
1/2)
=
2
(& P. (A)
^ l
=
_
Q(
£L , xp(^ ,x) x exp
x +
(^) a \
/
)
2i\
f
I.
(x
where the dependence on S
1
1
S„ ')
dx
de
A is due to the fact
defined previously depends on A. P
=
p e
(A)
Then, Pg
that the term
becomes
P(A) dA
(4.1) faCi
1 2tt
,x)x exp(-
Q( >0
52
)
S..
I* (x
dx
)
dA 4sexp(- T^l) £.A
d
^
i_i
IX
-
t
Q(^S,x) L
o
x
exp(--|)
Jo
(
~°^)-A
exp
2
~K
Io
A
(
x
Zp
dA
i
dx] d8
equality has been obtained
where the last
by interchanging
the order of integration.
This equation does not appear to
be readily simplif iable
If we take a closer
.
innermost integral in Equation 4.1, evaluated using I
v
exp(-p 2 t 2
(a t)
)
look at the
that integral might be
dt
v
-a 2p
where
Iv
kind
of
,F,
1
)
the special
exp( P
r(vU)
order
,F,
IS.
Tl
*p"l
F fJi=H +1 ( l
2
v+1 ,
the modified Bessel function
is
(•)
(X,,X 2 ;X 3
function
u
v,
Gamma
the
is
(•)
»
Up 2)
of the first
function,
and
the Confluent Hypergeometric function.
is
case of X
=0,
,
(O.X^jX^)
=
1
the
In
Confluent Hypergoemetric
.
that the integral in question
so
can be simplified in this case.
However, since the term Sn includes the integration factor A, we may be able to calculate and
express Equation
4
.
1
in
a
simpler form
using the
formula above for the case in
which the variable A could be
separated out in the term /Sn
for a given jammer waveform.
The same arguments apply to obtaining the probability of
error under the hypotheses H
.
Thus
Pe
can be shown to be
given by 'co
q( ^°
2
.
l|i°,x)x exp (--j-
)
(4.2)
O
z ? A Z (S^fs7o)-A 4 ,
Az
exp
.
Q
I
53
(x -^2-)
dA
dx d*
Using Equations 4.1 and 4.2, we can evaluate the performance of receiver in a Rayleigh fading environment from P
e
=
-T
(P
+
el
P
eo)
assuming that the two signals are equally likely to be sent.
54
DESCRIPTION OF GRAPHICAL RESULTS
V.
A.
GENERAL In this chapter,
the analytical results of the previous
chapters are now presented via
graphical means based on the
derived mathematical expressions for receiver probability of error The plots presented display
the receiver probability of
error (Pe) as a function of SNR for the various jammer waveforms previously considered for a set of JSR values. In each
order to
plot,
JSR
the case
provide the
basis for
=
has been
included in
comparisons of
the jammer
receiver performamce as it additive white Gaussian noise only interference.
relates to
effectiveness on the
B.
ASK (ON
-
OFF KEYING)
The graphical
the incoherent
results for
performamce are presented first. numerical evaluations of likely hypothesis,
ASK receiver
These plots correspond to
Equations 2.5 and 2.7
that is,
P(H,)
and P(H
)
for equally
are equal to
1/2.
The plot of Pe for ASK modulation is shown in Figure 6.7 as a function of SNR for fixed values of JSR, as specified in Equation 2.6
.
increasing SNR.
(0.25
in this
,
db of
Thus,
if JSR increases beyond a
figure),
Pe
a Pe of
io~ 5
In comparison,
at a JSR value of 0.0.
SNR to
jammer
increases with
From this figure, one can observe that 16.0
db of SNR is required to obtain i.e.
a
Figure 6.7 clearly shows the
'break point' phenomenon in which
certain value
using
obtain the same
Pe for a
without jammer, it
takes 23.5
JSR value
of 0.1.
in the presence of a jammer with a JSR value below the
55
.
we need a larger
break point,
same performance
JSR
value
operating without
a
However, in a jamming environment,
a
level of
jammer interference. above
receiver
a
break
the
of SNR that can produce Pe of
For
the case of
a
value of JSR
occurs at
produces
point
increases with increasing SNR. point.
obtain the
SNR in order to
which
Pe
there is no value
In fact, 10~
a
for a JSR above the break
;
ASK modulation,
the
which is approximately
break point 0.25,
as
obtained from Equations 2.8 and 2.9. Figure 6.8 shows the variable thresholding effect on the
jamming situation with JSR =
0.3 beyond the break point
=
Instead of using the
0.25).
Equation
fixed threshold as given by
variation of
the
2.5,
the
receiver
threshold
obtained from Equation 2.5 by changing the value of R reduce the
jamming effect
values as shown. the variation of
(JSR
restricted range
over a
,
can
of SNR
since as shown in Equation 2.13
However,
not significantly affect
the value R does
variation over a wide range of values of R does not result in a significant change in Pe the threshold
C.
value,
the
FSK WITH TONE JAMMER This section
presents graphical
results pertaining
to
with a single tone jammer acting against one of the two channels and a jammer consisting of two different tones acting against both chan-
jamming effects on FSK modulation
nels simultaneously.
Figure 6.9 corresponds to the performance of the optimum FSK receiver in which pt 3.12 is used
1/2.
Equation 3.10 or Equation
to evaluate performance with
jammer specified in can be
-
thought of
Figure 6.9 shows
Equation 3.11 as
a
.
This
and 'space'
'mark'
the near optimum
jammer waveform channel jamming.
similar result to that found
case except that the breakpoint occurs
56
in the ASK
at a higher value of
JSR than that
found for ASK.
JSR somewhere
between 0.5
This breakpoint
and 1.0 as
occurs at a
shown in
this plot.
From this figure it can be noted that 13.5 db of SNR is need to obtain a Pe of ]_0~ f° r a JSR value of 0.0, but the same Pe is obtained by increasing the SNR to 16.5 db for a JSR of 5
This demonstrates
0.1.
require
significant
a
relatively
that
SNR
to maintain
order
boost in
values
low JSR
a
certain desired Pe value. As shown in Figures 6.7 and
comparison of ASK and
6.9,
FSK modulation reveals that FSK
is
to jamming.
remembered
However
it must be
somewhat less vulnerable that the jammer
waveform n.(t) used in each case is different. The effect of the single tone jammer on the optimum FSK receiver is presented in Figure 6.10 which corresponds to j.
evaluation of
the
Equation
3.15
jamming on either the 'mark' or same effect
insofar as
is
jamming is
a Pe of iq -5 as
19.5 db of SNR
compared
to an SNR of
with Pe
6.9 for a JSR value of 0.3
24.5 db in Figure
concerned.
channel jamming only is eval-
Note that in Figure 6.10,
required to obtain
channel
channel has the
'space'
single tone
Therefore the effect of 'mark'
uated and plotted.
the
single
The
.
=
iq~
5
also
Figures
As expected,
single channel
than simultaneous
jamming is less effective
jamming of 'space' j
and 10 demonstrate the fact that
9
channels with
and 'mark'
a
near optimum
ammer The
effect
considered
of a
by changing
variable
1/2,
value of
the
receiver shown in Figure 6.6
threshold
.
q£
on FSK in the
For a value of
the simultaneous jamming of
'mark'
qC
will
be
modified other than
and 'space'
channel
results in a compensation of the other channel such that the
jamming effect remains the same as in In other words,
it
is
difficult
jamming effect by means of
a
the case of qC = 1/2. to reduce the near optimum
varying the threshold.
57
.
when
On the other hand, to
the receiver,
jamming effect
the
the threshold
adjusting
one channel jamming is applied
can
with increasing
be reduced
by
shown
in
SNR
as
Figure 6.11, part icularily for JSR of 1.0 and various values that the 'mark' of #( A moment's reflection will reveal .
channel jammer increases the output power level of the upper so that when the
envelope detector (see Figure 6.6)
the error increases.
signal is sent,
for this
under the assumption that the 'space' signal
type of error, has been
transmitted,
level of
the output
priate value of D.
Therefore,
'space'
reduced by
can be
Pe
of the multiplier
lowering the
by using
an appro-
£>(
FSK WITH FM JAMMER This section presents
FM sweep jammer
the effect of an
using sinusodal modulation on noncoherent FSK signaling. The several times during
the signal
jammer effectiveness was number
sweep the bandwidth
was designed to
FM jammer
of times
a bit
interval.
investigated as
the jammer
sweeps
occupied by
a
Thus the
function
over the
of the
band of
the
signal during one bit-time interval.
Figure 6.12 shows the result for one sweep of the jammer
per
interval.
bit
increasing
Figure
the sweeping
6.13
to two
shows
sweeps
result
the
per bit
of
interval.
These plots show that the FM sweep jammer can be more effective by
increasing the number
of times of
jammer sweeping
during a bit interval. This can be expected from the results
obtained in Equation obtain of 0.3,
a Pe of io~ 5
but
,
3.27
.
In Figure 6.12,
16.0 db of SNR is required for JSR value
in Figure 6.13 which corresponds
jammer sweeping,
in order to
the same Pe
to twice the
can be obtained by increasing
the SNR value to 19.5 db for the same JSR value of 0.3.
comparison to
the previous case
of FSK with
In
tone jamming,
for the same
jamming environment (i.e.,
Figures 6.9 and are required 10"
6.10 show that 24.5
respectively in
value of 0.3)
JSR
db and 19.5 db
order to get
the same
of SNR
Pe of
5 .
These
different requirements
of SNR
value for
various
jammer waveforms show that the FM sweep jammer can be effecas effective as the near optimum
tive but in general is not jammer.
Note that from a practical point of view, the added
complexity of candidate
FM jammer
for a
waveform may
replacement of
the
make it
an unlikely
near optimum
jammer.
However, one advantage the FM sweep jammer has over the near
optimum jammer is that the former a
large
bandwidth easily
than the latter in the case
can spread its power over
and therefore
is more
effective
of lack of exact information or
knowledge about the signal carrier frequency.
59
VI.
CONCLUSIONS
The familiar model in which not adequate
interference is
has analyzed
when jamming
or interference
the transmission environment.
signals are present in thesis
Gaussian noise is the total
effect
the
This
deterministic
of various
probability of error on binary
jammer waveforms in terms of
incoherent receivers operating in the presence of noise.
From the jammer point of view,
the goal is to cause the
maximum possible error to the various receivers while making efficient
its available
use of
jammer power).
waveform
carry the
with
fixed
it was proved that
made of
is
signal proportional to the difference
used to
(i.e.,
For coherent receivers,
optimum jammer
the
power
a
deterministic
of the binary signals
digital information.
This thesis
has
demonstrated that those optimum jammers derived for coherent receivers
perform their
function as
near optimum
jammers
satisfactorily against incoherent receivers. these nea-r optimum
Therefore, to be
jamming
jammers can be concluded
most attractive candidates
one of the of binary
incoherent
for efficient
communication systems.
An
optimum jammer has not been
derived or analyzed because the complexity of the expression for Pe makes it very difficult if not impossible
closed form.
jamming
to derive the optimum
jammer waveform in
Other jamming waveforms such as single channel
and FM
sweep
jamming
comparison to near optimum jammer ciency can be reduced by
showed its
inferiority
in
waveforms and their effi-
means of appropriate variations of
the receiver threshold.
60
.
s
d
(t)
C
=
Y
=
Figure 6.1
=
bias
S
s
=
In R
(t)
t
2
Sn(t)
S^Ct)] dt
R
;
Correlation Receiver for Binary Signal gnais
61
coswc
t
-A T/2N I
Figure 6.2
;(t)
)
=
R
Quadrature Receiver for ASK
sin WC (T -
=
(2An/N Q
t)
H>
0^
t
£T
r(t)
h(t)
r
cos W C (T
-
t)
0£ t£T
Figure' 6.3
Alternate Form of Quadrature Recei ver
62
.
"(t)
sin
=
Kt),
o<
t
t-'
c
(T - t)
Envelope
*t
Figure 6.4
Detector
V)
Incoherent Matched Filter Receiver.
h,(t)=sinW,(T-t)
Envelope Detector
-*
^
£u
t
^T
+ Rr rtt)
tyt) = sinW#(T-t)
H >< o
Envelope
^
Pe terror
Figure 6.5'
Incoherent Frequency Shift Keying (FSK) Receiver,
63
hl(t) = sinW,(T-t) ^ t
LT
Envelope Detector
2(i-a)
m
h (t)
=
sinWjT-t)
£:
t
£T
Envelope Detector zoc
Figure 6.6
Modified Incoherent FSK Receiver
64
i
r ,
o
•—
in CJ CO vn
OOo
Qd o o o O CO in w" X 0i X X a OS K "K Wco m en in 00 CO _;
II
i-4 •-»
II
II
II
—) —i —> -j
II
II
en en *-»
-»
II
—
c:
o
o < + X o >
•H J_>
«J f-i
a •a
o
CO <;
o V) (U
-i
E E •H X-l
a
o
u 55
1-1
3 •H
65
r
1
c o
•H iJ cd
rH
a
o
O
©
<
o
to
o o o qO c o d — — —«
<:
c O
••
o" ^o
II
II
•u
II
O
QoQ So o o o
0J
rjiJ -4 _! -)
M-l M-(
wx X £: .-JC/3
W
C/3 •CD
03
K c£ K a K ft 15 H D o <
o M
+
o d
H XJ a)
•H J-i
o
>
d
+-r I
3d
66
i—l
o d
—
u 3
»-or
CO
•H
>
o •^ CO
fid o o
2
ii
11
II
irt
i
oOO
o _^ <0 m II
II
II
ii
k£ft UK a:a wen en CO en en en CO
J-* — —
D o <
m— -
X
—
—i
c o
>
•H
iH
D T3
O
u in U*
u o V4
E=
E
S £ •H 4J
O t-i
u 3 00
67
o
CO
II
II
poo coo
^
C o
\
01
u 3 aO
•H
69
'
o ~m CO in oo O ao = LEGEND
ii
II
II
II
II
n
>
II
a: JSR C/l t/j
— c o
K/)(/)(/3lC0i
•H
D
U
O
»
X
>
cd
rH
3 T3
O
£ M CO
o 6
Pn
u
m o
tH —» —
i
D o <
»-»
it
,
II
(71
—
c o •H
+ X o >
JJ CJ
rH
a O T.
U O
<3J
n T3
°
en
Q X
cd
3 TJ
2: en
O
^ CJ
C <" 1
3 cr
that
UL,
is,
/$
>
oC
J
3
60,
Marcum Q- function can be computed from Q(
a
,
6
)
=
1
-
—
•UL
x
exp
'LL
74
(
—
- a)
f(x) dx
the
On the other hand if the lower limit LL is negative for some
value of q£ the case of
,
then there are two cases to be considered.
^
<
that is,
UL,
&
<
the Marcum
/ 360,
o( +
In
Q- function can be substituted with th^ value given by
QC a
,
B
)
=
1 -
—
46
exp
x
a
Finally in the case of
oLB
>
UL,
B
that is,
^
(/.
+
the value of Marcum Q-function can be computed from
Q(
a
,
3
)
=
1
/X a
(—
-
-
.2
a)
f(x) dx
a
Here it has been assumed that e
I Q (X)
does not
the function f(X)
impose the limitation of
the digital computer and that
defined by
computation on
the library functions for the
computation of e Ij(X) and its integration accuracy are available to the user.
75
with desired
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LIST OF REFERENCES
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Bukofzer, D. Performance of Optimum and Subopt imum Incoherent Di gital Communication Receivers in the Presence of Noise and Jamming, Final ReporT for Research Contract No ~~5156- 5 160° February 1984. ,
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A. Whalen, D. Academic Press, 1971.
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Srinath,
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and Rajasekaran, P. K., An Statistical Signal Processing WiTTh John Wiley and Sons, 1979 D.
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Ziemer,
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Part
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PT~
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INITIAL DISTRIBUTION LIST No
.
Copies
Naval Academy Library Chin Hae Republic or Korea
1
2.
Library, Code 0142 Naval Postgraduate School Monterey, California 93943
2
3.
Professor D. Bukofzer, Code 62Bh Naval Postgraduate School Monterey, California 93943
5
4.
Professor S. Jauregui, Code 62Ja Naval Postgraduate School Monterey, California 93943
2
5.
LT Joo, Hae-Yeon 976-31 13 Tong 3 Ban Dae-Lim 1 Dong, Yeong-Deung-Po Ku Seoul, Republic of Korea
2
6.
Defense Technical Information Center Cameron Station Alexandria, Virginia 22314
2
1.
,
77
13
3 7
211331 Thesis JT52 c.l
Joo
An analysis of jamming effects on noncoherent digital receivers.^-"-
