International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 5
Number 2
June 2011
183
1 INTRODUCTION
One of the main characteristics determining the ef-
fectiveness of a radio communication system is the
stability against disturbances [1,2]. It is character-
ized with the dependency of the fidelity of received
communications on the line energy parameters, al-
gorithms used to transmit information and statistical
characteristics of interferences [1,2]. With discrete
systems of connections, the error probability of dis-
tinguishing signals is used for fidelity assessment
[3,4].
The modern radio communication systems with
safety responsibility are required to guarantee that
the error-probability given will not exceed the pre-
liminary specified permissible value independently
of the variability activity of the channel. In essence
it means that the given quantity of the system func-
tioning is achieved thanks to the independency (par-
tial or complete) of the performance of the noise-
resistance from reasons causing the non-stationary
state of the channel of connection. In the theory of
automatic control this ability of the system to oppose
resist against the disturbing actions is known as in-
variance. In the systems of connection, the part of
disturbing effects is played by different disturbances
and noise-resistance is the feature of the system that
is invariant to them.
2 DEFINING THE CONDITIONS OF
INVARIANCE
For a radio channel, the typical situation is the one
where the performance of noise-resistance is deter-
mined by the presence of disturbances of several
classes (fluctuating, spectrum-concentrated, im-
pulse). The functional kind of the expression of the
error probability with receiving by elements depends
on the sets of signal parameters, the disturbances and
the interaction between them:
{ }
{ }
[ ]
222
;;
iji
GhhP
ζ
=
, (1)
where
2
i
h
and
2
ζ
h
express the mean statistical prop-
erties of the ratios between the energies of the i-th
signal variant and the j-th disturbance variant and
the white noise spectral density.
The part of parameters of interaction between the
signal and disturbances is played by set {
2
ij
G
},
i=1÷n , j=1÷
ζ
n
, average statistical values of the co-
efficients of the reciprocal differences in the fre-
quency-and-time area of their structures.
As the degree of the interaction between the use-
ful signal and the disturbance on the frequency-and-
time plane is analogous to their mutual correlation
function, it is suitable to assume the average statisti-
cal value of the mutual difference coefficient in the
position of interaction between them. This value is
expressed in the kind of:
An Invariance of the Performance of Noise-
Resistance of Spread Spetrum Signals
G. Cherneva & E. Dimkina
University of Transport, Sofia, Bulgaria
ABSTRACT: The paper is a study on the invariance of the performance of noise-resistance with respect to the
quasi-determined spectrum-concentrated interferences with transmitting of spread spectrum signals with a
fixed algorithm of processing.
184
( ) ( )
2
0
*
0
2
2
Σ=
∫
T
i
i
ij
dtttS
TP
KK
G
j
ζ
ζ
, (2)
where К
0
, К
ζ
, are the amplitude coefficients of the
signal and disturbance, Т=
τ
0
N is the signal length,
( )
tS
i
and
( )
t
j
*
ζ
Σ
are the complex functions of the i-
th signal and j-th disturbance,
( )
∫
=
T
ii
dtts
T
K
P
0
2
2
0
is the average power of the i-th
signal variant.
The conditions of the invariance of the
connection system are expressed in relation to a
certain class of disturbances and in dependence with
the metrics selected on the signal space.
If n(t) and
( )
t
ζ
are random realizations of fluctu-
ation noise {N} and quasi-determined interferences
{Ξ} respectively, then the performance of noise-
resistance is a function of both interferences:
( )
,,Ξ= NPP
. (3)
The system of connection is absolutely invariant
to
( )
t
ζ
, if :
)()0,(),( nPnPnP ==
ζ
(4)
is fulfilled.
When the noise-resistance characterization de-
pends on interferences Ξ to a certain extent, e.g.:
εζ
≤− ),()0,( nPnP
, (5)
then the system is relatively invariant (invariant to
ε), where ε presents the given distance between
),(
ζ
nP
and
)0,(nP
:
( )
0,),(max nPnP −=
ζε
ζ
(6)
3 STUDY ON THE INVARIANCE OF THE
PERFORMANCE OF NOISE-RESISTANCE IN
REGARD TO SPECTRUM-CONCENTRATED
INTERFERENCES WITH COHERENT
RECEIVING SSS.
Under the condition of the effect only of fluctuating
white noise {N} the noise-resistance of the system
is determined by the ratio of signal energy power
W
1
to the spectrum density of white noise power
2
0
ν
:
2
0
1
2
ν
W
h =
. (7)
With transmitting opposed signals and fixed ratio
signal/noise, the optimal operator of the receiver that
is to ensure maximum noise-resistance against the
interference {N} is the algorithm of coherent receiv-
ing with probability of error [3]:
( )
[ ]
21
2
1
hP Φ−=
, (8)
where Φ(.) is the integral function of distribution of
Cramp.
With complicating the noise situation in the
channel, when besides the fluctuating noise there are
also effects caused by spectrum-concentrated inter-
ferences
( )
t
ζ
, the probability of error is determined
with independence [2]:
( )
[ ]
е
hP 21
2
1
Φ−=
, (9)
where
e
h
considerably depends on the type of re-
ceiver and the frequency- and time properties of the
signals processed and the influencing disturbances.
For a receiver optimal for channels with fluctuation
noise and working with the influence of spectrum-
concentrated disturbances:
22
2
2
1
ij
e
Gh
h
h
ζ
+
=
(10)
It is function
( )
hP
of the probability of error
from parameter h that appears in the capacity of in-
variant of the system in relation to disturbance
( )
t
ζ
.
Taking into consideration dependencies (9) and (10),
it is obtained for it:
( )
+Φ−=
−
2
1
22
121
2
1
ij
GhhP
ζ
. (11)
The characteristic of noise-resistance obtained
depends of the effect of
( )
t
ζ
and
( )
2
ζ
hfР =
. There-
fore:
( ) ( )
0,, hPhP ≠
ζ
,
( ) ( )
tinhP
ζ
var≠
,
i.e. the condition of absolute invariance (4) to spec-
trum-concentrated disturbances has not been satis-
fied.
The Table 1 gives the values of the probability of
error calculated according to dependency (10) for
the cases when there are no concentrated disturb-
185
ances (
2
2
ij
Gh
ζ
=0) and when a disturbance of
2
2
10=
ζ
h
и
32
10
−
=
ij
G
is influencing.
From the analysis of the data in Table 1 it follows
that the maximum increase of the probability of er-
ror in area
2
10
−
≤Р
is
3
10.259,6
−
. According to
the condition of invariance (3) it follows that the rel-
ative invariance feature of noise-resistance up to
3
10.259,6
−
=
ε
is available in respect to spectrum-
concentrated interferences .
When the concentrated interferences are of uni-
form spectrums, the coefficient of reciprocal differ-
ence
2
ij
G
from the signal basis can be expressed as
[2]:
TF
G
i
ij
ρ
=
2
i=1,2, (12)
where, with given
( )
t
i
S
и
( )
t
j
Σ
,
ρ
is a constant
quantity located in the interval:
TFTF
i
≤≤≤
ζ
ρ
1
,
as the left limit of the interval is valid for a sinus-
like shape of disturbance. With
constF
i
F ==
, de-
pendency (10) takes the kind of:
FT
h
h
h
e
2
2
2
1
ζ
ρ
+
=
(13)
Table 1. Values of the probability of error
0.5
0.079
0.023
7.153·10
-3
2.339·10
-3
7.827·10
-4
2.66·10
-4
9.141·10
-5
3.167·10
-5
1.105·10
-5
3.872·10
-6
1.363·10
-6
4.817·10
-7
1.707·10
-7
6.066·10
-8
2.16·10
-8
0.5
0.089
0.028
9.759·10
-3
3.5·10
-3
1.284·10
-3
4.785·10
-4
1.802·10
-4
6.841·10
-5
2.614·10
-5
1.004·10
-5
3.872·10
-6
1.499·10
-6
5.818·10
-7
2.265·10
-7
8.834·10
-8
0
2
2
=
ij
Gh
ζ
10
2
2
.Gh
ij
=
ζ
P(h)
It is in Fig. 1 where the dependency of the proba-
bility of error
)
2
(hfP =
determined by dependency
(9) for different values of
FT
h
2
ζ
ρ
has been studied.
Under the conditions of influence of only a fluctuat-
ing noise (Curve 1), the probability of error is de-
termined only from
2
h
regardless of the shape of
signals that are transmitted. With
10
2
≥
FT
h
ζ
ρ
(curves 3 and 4), the efficiency of the coherent re-
ceiver under examination has been reducing consid-
erably.
10
P
h
2
10
3
10
2
10
-5
10
-4
10
-3
10
-2
10
-1
1
1
2
=
FT
h
ς
ρ
10
2
=
FT
h
ς
ρ
2
10
2
=
FT
h
ς
ρ
1
2
3
4
5
6
Fig.1. The dependency
)
2
(hfP =
for different cases
For the radio channels of decimeter range, the in-
tensity of the concentrated interferences s is charac-
terized by
4
2
1010 −=
ζ
h
. Hence, to provide invari-
ance in respect to concentrated interferences and to
guarantee the given noise-resistance level, it is nec-
essary to use complex signals with a basis size de-
pending on the ratio
2
ζ
h
.
The complicated noise background requires an
optimization of the circuit of the receiver and adap-
tation of its structure depending on the interfering
effects. In all known cases of systems designed with
considering the effect of fluctuating noise and spec-
trum- concentrated interferences [2], the expres-
sions of the probability of error depend monotonous-
ly on the value of the product for random j:
2
22
0
1
j
j
h
hG
j
j
ζ
ζ
δ
+
=
, (14)
so that
( )
j
h
e
h
δ
−= 1
22
. (15)
186
Taking into account (14) and (15), the probability
of error can be expressed as:
+
−Φ−=
2
1
2
2
0
2
1
121
2
1
ζ
ζ
h
Gh
hP
j
(16)
With an intensity of the spectrum-concentrated
interferences , such as
2
ζ
h
≥
10, it is obtained that :
( )
1
1
2
2
≈
+
ζ
ζ
h
h
and
( )
22
0
ζ
δ
hconstG
jj
=≤
.
Hence
2
e
h
does not depend on
2
ζ
h
and the opti-
mized receiver is invariant with regard to the influ-
encing spectrum-concentrated interferences unlike
the receiver of nature (8). Besides that, for signals
of sufficiently big bases
1<<≤
FT
j
ρ
δ
and
2
e
h
≈
2
h
has been provided, i.e. in practice, the noise-
resistance of that receiver does not differ from the
noise-resistance in channels only of fluctuating
noise. In Fig.1 what has been studied is the depend-
ency of the probability of error from
2
h
with two
values of coefficient
j
δ
- curve 5 (with
4
1
=
j
δ
) and
curve 6 (with
10
1
=
j
δ
).
4 CONCLUSIONS
The paper presents the obtained analytical depend-
encies between the size of the basis of the transmit-
ted complex signals and the possibilities of coherent
demodulators to compensate the effect of interfer-
ences. A coherent demodulator optimal in respect to
white noise and an optimized demodulator operating
under the conditions of the effect of white noise
spectrum-concentrated interferences have been
compared. The results obtained have shown that the
optimized receiver can keep a fixed level of proba-
bility of error with considerably smaller signal bases
providing an absolute invariance of the nature of
noise-resistance against spectrum-concentrated inter-
ferences.
REFERENCES
1. Proakis, J. 2001. Digital Communications. Mc Graw Hill
Series in El.Eng. Stephen W.
2. Haykin, S. 1994.Communication Systems. Willey & Sons,
USA.
3. Simon, M.K., J.K. Omura, R.A. Sholtz. 2001. Spread Spec-
trum Communications. Handbook, Hardcover.
4. Holmes, J.K. 1982. Coherent Spread Spectrum Systems.
Willey, New York.
