Cet article traite de la classification des modulations numériques linéaires. L'algorithme de classification proposé est basé sur les statistiques cycliques du second ordre et d'ordre supérieur du processus d'observation. Des exemples de simulations sont présentées pour montrer l'efficacité de la méthode proposée

IZZO, Luciano

MATTERA, Davide

I-Revues

A cyclostationarity-based classier
for digital linear modulations
Luciano Izzo, Davide Mattera
Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni
Universita degli Studi di Napoli Federico II
Via Claudio, 21, I-80125 Napoli, Italy
izzoluc@unina.it, mattera@unina.it
Resume { Cet article traite de la classication des modulations numeriques lineaires. L'algorithme de classication propose est
base sur les statistiques cycliques du second ordre et d'ordre superieur du processus d'observation. Des exemples de simulations
sont presentees pour montrer l'ecacite de la methode proposee.
Abstract { The paper deals with the classication of digital linear modulations. The classication algorithm is based on
second- and higher-order statistics of the observed process. Simulation examples are presented to show the eectiveness of the
proposed method.
1 Introduction
The problem of estimating the modulation format of an
incoming signal corrupted by noise and interference plays
an important role in several applications for both civil
and military purposes, such as monitoring, surveillance,
adoption of an appropriate warfare strategy, etc.
The modulation classication problem becomes very
complicated in the presence of a large number of alterna-
tive modulations. In such a case, it can be appropriate to
design the classier by resorting to a decision-tree struc-
ture that is based on sub-classiers that operate on sub-
sets of possible alternatives. Such an approach, allows
one to achieve computational savings with respect to the
one-step classier. Moreover, the design of each elemen-
tary classier is often very simple and the modularity of
the overall structure provides exibility and simplies its
updating aimed at improving the performances as well as
introducing new modulation formats among the possible
alternatives.
The classication problem under consideration is fre-
quently solved by extracting from the received signal some
discriminating features. Among the features considered
in the open literature, an outstanding position is held
by those based on the cyclostationarity properties of the
received signal, whose utilization in modulation format
recognition was rst proposed in [1] and, subsequently,
exploited in several papers [2]-[5]. The main advantage
of these features is that they are intrinsically tolerant to
the presence of noise and spectrally and temporally over-
lapping interfering signals, provided that the observation
time is suciently long and, moreover, at least a cycle fre-
quency of the signal to be classied is not shared with the
undesired signals.
In the present paper, we consider as features the cumu-
lants of the symbol sequence, which were already recog-
nized [6] to be eective for modulation classication. In
the paper, however, the considered features are estimated
by exploiting the second- and higher-order cyclostationar-
ity of the received signal. The proposed classier operates
on linear memoryless modulations and turns out to be
eective in severe noise and interference conditions; more-
over, it is exible and adequate to work in a decision-tree
structure. Therefore, it can be utilized as a preliminary
classier for partitioning various digital modulation for-
mats into subclasses (e.g., for discriminating between one-
and two-dimensional signal constellations), but may also
be adopted within each subclass to determine the exact
modulation format.
2 The proposed classier
The down-converted received signal can be written as:
r(t) = As(t) + i(t)
= A exp[j(+ 2f
0
t)]
+1
X
`= 1
a
`
p(t  `T + t
0
) + i(t) (1)
where A  0 represents the unknown (but nonrandom)
channel attenuation, p(t) is the known baseband pulse, T
is the known symbol period, f
0
is the known residual car-
rier frequency or frequency oset, t
0
and  are unknown
(but nonrandom) and fa
`
g is a complex-valued zero-mean
and i.i.d. symbol sequence normalized so that its variance
is unity. The zero-mean additive disturbance i(t) includes
thermal noise and possible interfering signals and is in-
dependent of the signal s(t), which is the signal to be
classied.
Let us dene s(n)
4
= s(nT
s
) with T
s
the sampling pe-
riod. Under the above-mentioned assumptions, it can be
shown [7] that the N th-order temporal cumulant function
of s(n) is given by
C
s
(n;m)
N
4
= cum[s
()
1
(n+m
1
); s
()
2
(n+m
2
); : : : ;
s
()
N 1
(n+m
N 1
); s
()
N
(n)]
= c
N;I
exp[j(
0
+ 2
0
(N   2I)n)]
1
X
`= 1
g(nT
s
  `T ;m)
(2)
where the superscript ()
h
(h = 1; : : : ; N ) denotes op-
tional conjugation, c
N;I
4
= cum[a
()
1
`
; a
()
2
`
; : : : ; a
()
N
`
] with
I the total number of conjugations, m
4
= (m
1
;m
2
; : : : ;
m
N 1
), 
0
= f
0
T
s
,
g(t;m)
4
= p
()
N
(t+ t
0
)
N 1
Y
i=1
p
()
i
(t+ t
0
+m
i
T
s
) (3)

0
4
= (N   2I)+ 2
0
[()
1
m
1
+ ()
2
m
2
+ : : :
+()
N 1
m
N 1
] (4)
where ()
h
is 1 in the absence and  1 in the presence of
conjugation on the hth term.
By expanding in Fourier series the term
P
1
`= 1
g(t  
`T ;m), it can be shown that the reduced-dimension cyclic
temporal cumulant function of s(n) is given by
C

s
(m)
N
4
= hC
s
(n;m)
N
exp( j2n)i = c
N;I
exp(j
0
)

1
X
`= 1
c
(N)
p
(`;m) exp(j2`t
0
=T )
~
[
`T s
T
+ 
0
(N   2I) ]
(5)
where hi denotes innite-time average,  is the cycle fre-
quency,
c
(N)
p
(`;m)
4
=
1
T
Z
+1
 1
p
()
N
(t)
N 1
Y
h=1
p
()
h
(t +m
h
T
s
) exp

 j2
`
T
t

dt
(6)
and
~
(t) = 1 for t integer and
~
(t) = 0 for t noninteger.
Let us assume that the sampling period T
s
is chosen
so that it exists at least one discrete-time cycle frequency
kT
s
T
+ 
0
(N   2I) corresponding to an unique continuous-
time cycle frequency. Then, from (5) it follows that
jC
kT
s
T
+
0
(N 2I)
s
(m)
N
j = jc
N;I
jjc
(N)
p
(k;m)j 8k 2 K
(7)
where K denotes the nonempty set
K
4
=

k :
`T s
T
6=

kT
s
T

mod1
8 ` 2 L; ` 6= k

(8)
with
L
4
=
n
` : c
(N)
p
(`;m) 6= 0
o
: (9)
With reference to the received signal r(n), the additivity
property of the cumulants leads to
C
kT
s
T
+
0
(N 2I)
r
(m)
N
= A
N
C
kT
s
T
+
0
(N 2I)
s
(m)
N
+ C
kT
s
T
+
0
(N 2I)
i
(m)
N
= A
N
C
kT
s
T
+
0
(N 2I)
s
(m)
N
8k 2 K  K
i
(10)
where the last equality follows from the denition of K
i
as the set of values of k such that kT
s
=T is an N th-order
cycle frequency of the sampled version of i(t). Therefore,
from (7) and (10) it follows that
jc
N;I
j =
jC
kT
s
T
+
0
(N 2I)
r
(m)
N
j
A
N
jc
(N)
p
(k;m)j
8k 2 K  K
i
: (11)
Let us note that, since jc
2;1
j = 1, from (11), specialized
for N = 2 and I = 1, it follows that
A
2
=
jC
kT
s
T
r
(m)
2
j
jc
(2)
p
(k;m)j
8k 2 K  K
i
: (12)
Such an equation allows one to estimate the unknown
channel attenuation A by exploiting only second-order
cyclic statistics.
Equations (11) and (12) lead to consistent estimators
of jc
N;I
j. Dierent estimators can be obtained accord-
ing to dierent choices for k, m, I, and N and dierent
estimators for C
kT
s
T
+
0
(N 2I)
r
(m)
N
. Moreover, although
not explicitly indicated, both jC
kT
s
T
+
0
(N 2I)
r
(m)
N
j and
jc
(N)
p
(k;m)j also depend on the position of each optional
conjugation but for m = 0. Finally, note that for N  6
estimators of the cyclic cumulantC
kT
s
T
+
0
(N 2I)
r
(m)
N
can
be realized only on the basis of knowledge of the lower-
order cycle frequencies of the overall disturbance.
Equations (11) and (12) constitute a useful classica-
tion method for digital linear modulations, in that dier-
ent signal constellations exhibit dierent theoretical val-
ues of jc
N;I
j. For example, the proposed algorithm, uti-
lized assuming N = 2 and I = 0, that is, by exploiting
only the second-order cyclostationarity, is able to discrim-
inate one-dimensional from two-dimensional signal con-
stellations with equally likely symbols. Moreover, in the
case of N = 2 and I = 0, if the pulse p(t) is real and the
same pair of values (k;m) is assumed in both (11) and
(12), it results that knowledge of p(t) is not required.
The discrimination between one- and two-dimensional
constellations can be utilized as a rst step in a decision-
tree structure for the overall classication. In fact, the
discrimination inside the set of one-dimensional constel-
lations and inside the set of two-dimensional constella-
tions can be obtained by utilizing jc
N;I
j for N  4. Note
that often N = 4 is an adequate choice. However, at
least N = J is required to discriminate J-PSK from M -
PSK with M > J (theoretically, jc
J;0
j = 1 for J-PSK and
c
J;0
= 0 for M -PSK).
The implementation of the proposed classier requires
the comparison of the estimated feature (that is, an esti-
mate of jc
N;I
j) with a set of N
c
  1 thresholds, where N
c
is the number of classes in which the digital modulated
signals have to be partitioned.
Since the estimate of C
kT
s
T
+
0
(N 2I)
r
(m)
N
can be mod-
eled as a Gaussian-distributed random variable [8], by fur-
ther assuming that the channel attenuation estimate is
practically error-free, the estimate of the discriminating
feature provided by (11) can be reasonably modeled as
a Rice-distributed random variable, whose variance 
2
is
unknown, but is largely independent of the considered sig-
nal constellation [6]. The mean value, vice versa, depends
on the considered constellation.
The threshold (
2
) and the probability of correct clas-
sication P (
2
) are both functions of 
2
that can be nu-
merically evaluated. Consequently, the threshold setting
can be made by xing a desired value P
d
for the proba-
bility of correct classication, and determining the value

2
d
= P
 1
(P
d
), and, nally, the threshold 
d
= (
2
d
).
Note that the selected threshold is certainly satisfactory
for values of probability of correct classication quite close
to P
d
. Finally, it is useful to emphasize that both the
probability of correct classication and the threshold set-
ting depend on the known a priori probability distribution
of the candidate signal constellations.
3 Simulation results
To substantiate the eectiveness of the proposed classi-
cation algorithm, simulation experiments were carried
out. All experiments consider a scenario with three in-
dependent signals, in which the signal to be classied is
contaminated by a single interfering signal and thermal
noise.
In all experiments, the signal to be classied is linearly
modulated with 
0
= 0, T=T
s
= 4 and a raised-cosine-
shaped pulse p(t) using a 50 % excess bandwidth. The
interfering signal is a BPSK signal with the same pulse
p(t) and the same carrier frequency, but with a symbol
period 1.4 times larger. Therefore, in (11) and (12) it is
chosen k = 1 that leads to work with a cycle frequency not
shared with the interfering signal. The selected lag vector
is m = 0. The inband signal-to-noise ratio (SNR) is set
to 0 dB. The input parameter P
d
for the threshold setting
procedure (see Section 2) is set to 0.99 and the candidate
signal constellations are assumed to be equally likely. All
the results are based on 200 Monte Carlo trials for each
candidate signal constellation.
In the rst experiment, we consider the set f BPSK,
8-PSK g of alternative constellations. In such a case, the
second-order cyclostationarity is sucient to discriminate,
in that jc
2;0
j = 1 for BPSK and c
2;0
= 0 for 8-PSK. The
threshold resulting from the setting procedure is 0:58.
In Fig. 1, the probability of correct classication is
reported as a function of the signal-to-interference ratio
(SIR) and for dierent values of the number of received
symbols. The interference-rejection capability of the me-
thod is evident. Satisfactory performances are already
obtained with 100 symbols also in severe conditions of
noise and interference (SIR=SNR=0 dB). A number of
1000 symbols allows one to well discriminate also in very
severe interference situations (e.g., SIR= -8dB).
In the second experiment, we consider the case in which
the set of constellations to be classied is f BPSK, 4-PSK,
8-PSK g. In such a case, to discriminate 4-PSK from 8-
PSK it is not appropriate the second-order statistics c
20
which is zero for both constellations. Then, the proposed
strategy considers a rst step in which the estimate of
jc
2;0
j leads to the two sets f BPSK g and f 4-PSK , 8-
−10 −8 −6 −4 −2 0 2 4 6 8 10
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Signal−to−interference ratio (dB)
Pr
ob
ab
ilit
y 
of
 c
or
re
ct
 c
la
ss
ific
at
io
n
SNR = 0 dB
100 symbols
1000 symbols
10000 symbols
Fig. 1: The performance of the proposed classier oper-
ating on the set f BPSK , 8-PSK g.
PSK g and a second step where the estimate of jc
4;0
j is
used to discriminate between 4-PSK and 8-PSK. The se-
lected thresholds are 0.61 in the rst step and 0.58 in the
second step. Let us note that the more expensive esti-
mation of the fourth-order cyclic cumulant is performed
only after the separation (by the rst decision step) of
BPSK from f4-PSK , 8-PSK g. Both the computational
complexity reduction and the probability of correct classi-
cation increase as the probability of occurrence of BPSK
increases.
−5 0 5 10 15 20 25 30
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Signal−to−interference ratio (dB)
Pr
ob
ab
ilit
y 
of
 c
or
re
ct
 c
la
ss
ific
at
io
n
SNR = 0 dB
100 symbols
1000 symbols
10000 symbols
Fig. 2: The performance of the proposed classier oper-
ating on the set f BPSK , 4-PSK , 8-PSK g.
Figure 2 presents the results of the second experiment.
In such a case, 100 symbols are sucient to work well in
a severe noise environment (SNR=0dB) only for high val-
ues of SIR. However, 1000 symbols provide a very good
performance in severe conditions of both noise and inter-
ference.
In the last experiment, we consider the case in which the
set of constellations to be classied is f BPSK, 4-PSK, 8-
PSK, 16-QAM g. In such a situation, at rst the BPSK
is discriminated from f 4-PSK , 8-PSK, 16-QAM g by the
estimate of jc
2;0
j (theoretically, jc
2;0
j = 1 for BPSK and
c2;0
= 0 for all others). Then, in a second step, we utilize
the estimate of jc
4;0
j to discriminate inside the set f4-PSK
, 8-PSK, 16-QAM g, in that theoretically jc
4;0
j = 1 for 4-
PSK, c
4;0
= 0 for 8-PSK, and, nally, jc
4;0
j = 0:68 for
16-QAM. The selected thresholds are 0.63 in the rst step
and 0.32 and 0.84 in the second step.
−5 0 5 10 15 20 25
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Signal−to−interference ratio (dB)
Pr
ob
ab
ilit
y 
of
 c
or
re
ct
 c
la
ss
ific
at
io
n
SNR = 0 dB
100 symbols
1000 symbols
10000 symbols
Fig. 3: The performance of the proposed classier oper-
ating on the set f BPSK , 4-PSK , 8-PSK, 16-QAM g.
Figure 3 shows the results of the third experiment. They
conrm that also in severe conditions of noise and interfer-
ence a few thousand symbols are required to obtain very
good performances.
4 Conclusions
A new method for classifying linear memoryless digital
modulations has been presented. The considered discrim-
inating features are the cumulants of the symbol sequence.
The utilization of these features leads to a simple and ex-
ible design of the overall classier for the considered set of
alternative signal constellations. The main contribution
of the paper lies in the method for estimating the consid-
ered features that exploits the second- and higher-order
cyclostationarity properties of the received signals. As a
consequence, the proposed classier turns out to be eec-
tive in severe noise and interference environments, as is
conrmed by the results of the reported simulation exper-
iments.
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A cyclostationarity-based classifier for digital linear modulations

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