Table Of ContentSeparation of Signals Consisting of Amplitude and
Instantaneous Frequency RRC Pulses Using SNR
Uniform Training
Mohammad Bari and Milosˇ Doroslovacˇki
Department of Electrical and Computer Engineering
The George Washington University
6
Washington, DC, USA
1
mustafa,doroslov @gwu.edu
0
{ }
2
n
a Abstract—This work presents sample mean and sample vari- implementationlesschallenging.Furthermore,weusesupport
J
ancebasedfeaturesthatdistinguishcontinuousphaseFSKfrom vectormachines(SVM)basedclassifier. Ingeneral,classifiers
0 QAM and PSK modulations. Root raised cosine pulses are used
require the estimate of signal to noise ratio (SNR) to train
3 for signal generation. Support vector machines are employed
for signals separation. They are trained for only one value of itself. Therefore, classifiers are to be trained for every value
] SNR and used to classify the signals from a widerange of SNR. of SNR. We, on the other hand, train SVM for only one
T
A priori information about carrier amplitude, carrier phase, value of SNR. The classification threshold computed for one
.I carrier offset, roll-off factor and initial symbol phase is relaxed. value of SNR is then used to classify the signals having
s Effectiveness of the method is tested by observing the joint
a large range of SNR. Low number of realizations (50) is
c
effects of AWGN, carrier offset, lack of symbol and sampling
[ used for training. To the best of our knowledge there is
synchronization, and fast fading.
1 Index Terms—Digital modulation classification, root raised no published work on how to separate CPFSK signals from
v cosine pulses, support vector machines, training. QAM and PSK modulations where all modulations use RRC
7 shaped pulses, and where training is as short as presented in
5 I. INTRODUCTION this work. Standard deviation based features presented in [5]
0 Designing a classification algorithm to solve the Digital separatefrequencymodulatedsignalsfromlinearmodulations
0
Modulation Classification (DMC) problem is a challenging for rectangular shaped pulses.
0
undertaking. Three main issues should be considered. First,
.
2
the signal of interest propagates through a channel which Our choice of the features is motivated by the work pre-
0
can affect the signal by distorting it with noise, Doppler sented in [6], where a simple frequencyestimator is proposed
6
1 shift and multipath fading. Second, one or more of the main usingthe phaseof a productof consecutivecomplexsamples.
: parameters shaping the signal of interest can be unknown. In ourcase, the goalis to distinguishbetween linearand non-
v
Third, the complexity of the classification algorithm can be linear digital modulationsand instead of using the phase of a
i
X high. This work concentrates on the simple solution to the product of consecutive complex samples, we use the sample
r preprocessing part in which continuous phase frequency shift mean and sample variance of imaginary part of the product
a
keying (CPFSK) modulation is separated from linear modu- of consecutive complex samples. For rectangular pulses, the
lations of phase shift keying (PSK) and quadrature amplitude samplemeanoftheimaginarypartoftheproductisdiscussed
modulation (QAM). All modulationsuse RRC shaped pulses. in specific DMC scenarios in [7]-[9] where only binary FSK
Wavelet-based feature is presented in [1] for distinguishing (BFSK) modulation is separated from linear modulations.
frequency shift keying (FSK) from PSK having rectangular These pulses are not used in majority of modern systems.
shaped pulses. Approximate entropy is exploited in [2]-[4] to Additionally,insteadofSVM,thresholdbasedclassificationis
distinguishwithin the class of CPFSK havingnon-rectangular discussed in those papers. Since a lot of a priori information
instantaneous frequency pulses. Neural network classifier is onsignalparametersis neededto calculatethresholds,the ap-
used in [3] and it is trained on 500 realizations. High training proachisoflesspracticalinterest.In[10][11],theexactknowl-
requirement coupled with the high number of samples per edgeof the SNR of the receivedsignalis assumed.Reference
realization (8000) make it challenging for practical imple- [12] surveys SNR estimation techniques. The knowledge of
mentation, especially in real-time. In contrast, our method the value of SNR is relaxed by 3 dB in [13].
workswellwhenFSKcarriersareinseparableinthefrequency
domain, and when the number of samples taken in a symbol In sections II, III and IV the baseband discrete-time signal
period is low, which ends up in processing fewer number modelusedinthispaper,theproposedfeaturesandsimulation
of samples in a realization. This in turn makes real-time experiments are discussed, respectively.
II. SIGNAL MODEL where * is the complex conjugate operator. From now on, let
us assume that there is no fading, i.e., α[k] = 1, ψ[k] = 0.
Considera scenariowhere spectrumofa signalis observed
Now,applyingw[k]onnoiselessQAMandPSKsignalsyields
and its center is estimated (not considered in this paper).
Thenthe signal’sspectrumistranslatedtobearoundacertain
+∞
sdiegsniraeldshnoourlmdabliezecdlafsrseifiquedenacsy,CsPayFS0Koorrπli/n2e.arTlyhemoobdtualianteedd w[k]|v[k]≡0 = a2nej∆′Pn[k]Pn−[k]+
n=X−∞
one. Let complex baseband continuous-time received signal (6)
+∞ +∞
be a a ej[θn−θm+∆′]P [k]P−[k]
s(t)=x(t−t0)ej(∆t+θc)α(t)ejψ(t)+v(t) (1) n=X−∞mX=−∞ n m n m
wherex(t)isthetransmitted-signalpartofthereceivedsignal,
whereP [k]P−[k]=p(kT t nT)p(kT T t mT).
t0isthetimedelay,θc istheinitialphaseuniformlydistributed n m s− 0− s− s− 0−
over [0, 2π), ∆ is the carrier offset, (α(t), ψ(t)) are the Similarly, applying w[k] on noiseless FSK signals yields
amplitude and phase of the multiplicative noise used here
to model fading. α(t) is assumed to be a Rayleigh random j[ kTsR−t0 ω′(ρ)dρ+∆′]
variable, while ψ(t) is independent of α(t) and uniformly w[k]v[k]≡0 =e kTs−Ts−t0 Ts
distributed over [0,2π). v(t) is assumed to be zero-mean | (7)
bcoanmdpwleixdthnoBiseofatnhde rte0ceisivetrim’sefildteerlaiys.iWn egenasesrualmleargtheartththane =ejPn bnkTsk−TsRT−s−t0t0q(ρ−nT)dρ+j∆=′ ejPn bnQn[k]+j∆′
thebandwidthofthetransmittedsignalandthepowerspectral
density of v(t) is constant within the filter bandwidth. In the where Q [k]= kTs−t0 q(ρ nT)dρ.
caseofQAMandPSKmodulationsthetransmitted-signalpart Let usnconsideRrkTths−eTms−eat0ns of−imaginary part of w[k] in (6)
of the received signal is and (7) for linear and BFSK modulations, respectively. For
equiprobable constellation points of each modulation, mean
+∞
x(t)= a ejθnp(t nT) (2) of imaginary part of w[k] for 16-QAM, BPSK, 4-PSK and
n
− 8-PSK signals is given by
n=X−∞
where(a , θ )aretheamplitudeandphaseofthetransmitted
n n E[Im(w[k]v[k]≡0)]=E[Im(w[k])]
symbol, p(t) is the pulse shape function and T is the symbol |
period. In the case of FSK modulation =sin(∆′) +∞ P [k]P−[k]E[a2] (8)
n n n
n=X−∞
t
jRω(ρ)dρ
x(t)=e 0 (3)
where 16-QAM has constellation points anejθn k/√10+
where ω(ρ) = +∞ b q(ρ nT) is the instantaneous jl/√10; k, l = 3, 1,+1,+3 , BPSK’s p∈ha{ses θ
n=−∞ n − − − } n ∈
frequency of thePmodulated signal where q(ρ) defines instan- 0, π , 4-PSK’s phases θn (2n+1)π/4; n = 0,1,2,3 ,
{ } ∈ { }
taneousfrequencypulse shape and b 1,+1 for BFSK. and 8-PSK’s phases θ (2n + 1)π/8; n = 0,1,...,7 .
n n
∈{− } ∈ { }
Notethat(3)modelscontinuousphaseFSK,whichisofhigher Constellation points for 16-QAM, BPSK, 4-PSK and 8-PSK
practical interest than non-continuousphase FSK used in [?]. are chosen such that the average power is unity. Mean of
Now, we assume that we are sampling with period T = imaginary part of w[k] in (7) for BFSK signal is
s
1/(2B). Symbol period, T, is given by T = N T + εT
s s s
where N is the numberof sample periodsper symbolperiod ∞
s
′
and ε [0,1). The complex baseband discrete-time received E[Im(w[k]v[k]≡0)]=E[Im(w[k])]=sin(∆) cos(Qm[k]).
signal∈is | m=Y−∞
(9)
s[k]=x(kT t )ej(∆′k+θc)α[k]ejψ[k]+v[k]=s(t) (4) Next,letusconsiderthevariancesofimaginarypartofw[k]
s− 0 |t=kTs
in (6) for QAM and PSK modulations, and in (7) for BFSK
where ∆′ = ∆T , α[k] = α(kT ), ψ[k] = ψ(kT ) and v[k]
s s s modulation. The variance for BPSK is
is complexcircular AWGN having zero mean. Note that t =
0
k0Ts +ε0Ts where k0 is the integer part and ε0 [0,1) is +∞
the fractional part of the time delay t0 measured in∈sampling VAR[Im(w[k]|v[k]≡0)]=sin2(∆′) ( Pm[k]Pm−[k])2+
periods as time units. (cid:2)mX=−∞
(10)
+∞ +∞ +∞
III. PROPOSED FEATURES P2[k] (P−[k])2 2 (P [k]P−[k])2 .
m m − m m
Motivated by [6] we consider mX=−∞ mX=−∞ mX=−∞ (cid:3)
∗
w[k]=s[k]s [k 1] (5) For QAM, 4-PSK and 8-PSK modulations, whose signal
−
constellations are invariant to π/2 rotation, the variance is IV. SIMULATIONS AND DISCUSSION
VAR[Im(w[k]|v[k]≡0)]=sin2(∆′)(cid:20)(E[a40]−2E2[a20]) In this section, simulation experiments are presented illus-
trating the classification performanceof the proposedfeatures
+∞ +∞
1 for RRC shaped pulses. The performance for the proposed
(P [k]P−[k])2+E2[a2]( P [k]P−[k])2+ E2[a2]
m m 0 m m (cid:21) 2 0 × features is also compared to the performance of a classifier
mX=−∞ mX=−∞
using the wavelet-based feature for distinguishing PSK from
+∞ +∞ +∞
P2[k] (P−[k])2 ( P [k]P−[k])2 . FSK in [1]. We modify this approach by incorporating SVM
(cid:20) m m − m m (cid:21) in classification [14]. The number of training data used to
mX=−∞ mX=−∞ mX=−∞
(11) constructtheclassifiersforthetwoclassesis50realizationsof
eachmodulation.Forwavelet-basedapproach,SVMistrained
The variance of Im(w[k]) in (7) for BFSK modulation is
for every value of SNR. For the proposed features, SVM is
1 1 +∞ trained for only 10 dB. The corresponding SVM threshold is
VAR[Im(w[k]|v[k]≡0)]=2−2 cos(2Qm[k])+ thenusedforclassifyingsignalshavingseveralvaluesofSNR.
m=Y−∞ (12) The performance of the proposed features classified by SVM
+∞ +∞
that is trained for only one value of SNR is then comparedto
sin2(∆′) cos(2Q [k]) cos2(Q [k]) .
m − m the performance of the proposed features classified by SVM
(cid:0)m=Y−∞ m=Y−∞ (cid:1)
that is trained for every value of SNR.
For rectangularpulsesp(t) and q(t) with supporton [0,T),
Error performance of the methods is compared in the
amplitudes 1 and β′/T , respectively, unit average power
s presenceofadditivewhiteGaussiannoise(AWGN),unknown
signal constellations, and ǫ = ǫ = 0, E[Im(w[k])] in (8)
0 carrier offset, asynchronous sampling and fast fading. For
and (9) for linear and BFSK modulations, respectively, are
both the methods there are 70000 modulated signals, 10000
1 ′ signalsforeachmodulation(16-QAM,BPSK,4-PSK,8-PSK,
E[Im(w[k]|v[k]≡0)]=(1− Ns)sin(∆) (13) BFSK, 4-FSK and 8-FSK). There are 600 symbols in one
E[Im(w[k]v[k]≡0)]=cos(β′)sin(∆′). (14) realization. Discrete-time signals are obtained by taking only
| 2 samples per symbol period (N =2). Carrier offset, ∆′, and
For ∆′ = π/2, it is 1 1/N for linear modulations s
and cos(β′) for BFSK mod−ulations. Therefore, the mean of raorell-uonffifoorfmRlyRCdisptruilbsuetsedarienfi[γxe′d fπor/2a0,reγa′li+zatπio/n20a]n,dwhtheerye
Im(w[k]) at ∆′ = π/2 can be used to distinguish be- γ′ 0, π/2 and in k/10−; k = 1, 2, 3,...., 10 ,
tween the linear and the BFSK modulations. Similarly, the ∈ { } { }
respectively.Thesymbolperiodandtimedelayarenon-integer
VAR[Im(w[k]|v[k]≡0)] in (10) for BPSK, in (11) for 4-PSK, multiples of sampling period. This results in asynchronicity
8-PSK and 16-QAM, and in (12) for BFSK modulations,
between sampling instants and symbol period. ε and ε are
respectively, are 0
both uniformly distributed in [0,1) and remain unchanged for
VAR[Im(w[k]|v[k]≡0)]=N1s sin2(∆′)=(cid:26)0N1,s,∆∆′=′=π0/2 (15) adipstarritbiuctueldarirnea{l0iz,a1t,io2n,,..t.h,⌈aNt iss,+εǫ6=⌉−0,1ε}0.6=Fo0r. kth0eisfausntiffaodrminlgy,
α[k]eiψ[k] is such that its autocorrelation magnitude remains
1 1
VAR[Im(w[k]|v[k]≡0)]=(1− Ns)sin2(∆′)(E[a40]−1)+2Ns oarvoeurn1d/2fooufrtthimeefsadminogreatvhearnagNe p=ow2eirnfoourr9sismamulpalteios,nsw.hαi[ckh]iiss
s
1 ,∆′ =0 chosen to have unit mean square value, that is, E[α2[k]]=1.
=(cid:26)2N1s +(1 1 )(E[a4] 1),∆′ =π/2 (16) For optimal results of wavelet/SVM approach, the signals’
2Ns − Ns 0 −
VAR[Im(w[k]|v[k]≡0)]=(1−sin2(∆′))sin2(β′) fsoprecthtreacaorentpinlaucoeudsawroauvneldetnotrramnsafloizremdifsre2qaunednctyheπH/2a.aTrhweasvcealleet
sin2(β′), ∆′=0 (17)
= ′ is used as the mother wavelet. The length of median filter is
(cid:26)0, ∆=π/2 2 for wavelet-based method.
where E[a4] is 1.32 for 16-QAM and it is 1 for 4-PSK and First, the empirical evidence is discussed to prove the
0
8-PSK modulations. For β′ such that sin2(β′) > 1/(2N ) at usefulnessofthefeaturesforRRCpulsesinthejointpresence
s
∆′ =0, the variance of the BFSK modulations is larger than of AWGN, carrier offset, asynchronicity and fast fading.
the variance of linear modulations. On the other hand, the Figure 1 shows the difference in the closest sample means
varianceof the BFSK modulationis smaller than the variance and sample variances of Im(w[k]) between FSK and linear
of linear modulations at ∆′ = π/2. Therefore, the sample modulations. “h=x VAR (MEAN), P8-F8” correspondsto the
variances of Im(w[k]) at ∆′ = 0 and ∆′ = π/2 can be sample variance (mean) of Im(w[k]) for 8-PSK minus that
jointly used together with the sample mean of Im(w[k]) at of 8-FSK with h=x for carrier frequency centered at π/2.
∆′ =π/2todistinguishthelinearmodulationsfromtheBFSK “h=x VAR 0, F8-Q16” corresponds to the sample variance
modulation in the case of rectangular pulses. Furthermore, it of Im(w[k]) for 8-FSK with h=x minus that of 16-QAM for
can be shown that the features can be used to separate L-ary- carrier frequency centered at 0. Pm, Qm and Fm represent
FSK from linear modulations. L-ary PSK, QAM and FSK, respectively. It can be seen that
andP isthereceivedsignal’saveragepower.Estimatingnoise
S r
RE 0.5 power is a classical problem and it is outside the scope of
U
AT 0.45 this paper. A survey of SNR estimation techniques is carried
E
F out in [12]. It is worth mentioning that from the spectrum
F 0.4
N O h=1 VAR 0, F8−Q16 h=1 VAR, P8−F8 of BFSK signal the existence of two carrier frequencies is
EA 0.35 not observable even for h = 3/4. It is clear from the figure
M
WEEN 0.3 h=3/4 VAR, P8−F8 vthaalutethoefpSeNrfRormisacnocmepoafrparbolepotosetdhafetawtuhreersetrSaVinMedisfotrraoinnleydofnoer
T 0.25
E every value of SNR, espacially for SNR>4 dB. Based on the
B
CE 0.2 SNRofthesignals,SVMcreatesaclassificationthreshold.For
N
E h=3/4 VAR 0, F8−Q16 h=1/4 MEAN, F2−P2 separablecase,SVMtendstomaximizethethresholddistance
R 0.15
FE from the closest samples of the two classes in feature space.
F
DI 0.1 The distance increases as the SNR increases and hence the
T
S signals can be classified easily for higher values of SNR and
LE 0.05 h=1/2 MEAN, F2−P2 h=1/2 VAR, P8−F8
L vice versa. The simulation results show that the classification
A
M 0
S −5 0 5 10 15 20 25 30 threshold that SVM generates at 10 dB is good enough to
SNR [dB] classify the signals for the adjacent and higher values of
SNR. Though, SVM are trained for every value of SNR,
Fig. 1. Smallest difference in sample means and sample variances of wavelet/SVM simply fails to separate signals for small N .
Im(w[k])between FSKandlinearmodulations. s
Notethattheprobabilityoferrorforboththemethodsdecrease
by increasing N or N .
s
0.5
TO h=1/4 V. CONCLUSION
0.45 TO h=1/2
TO h=3/4 In this paper we propose simple and yet robust features to
0.4 h=1/4 distinguish CPFSK modulations from QAM and PSK modu-
h=1/2
0.35 h=3/4 lations that use RRC pulses in the joint presence of AWGN,
WAV h=1/4
carrier offset, lack of synchronization and fast fading. The
0.3 WAV h=1/2
WAV h=3/4 features are based on the sample mean and sample variance
Pe 0.25 of imaginary part of the product of two consecutive signal
values. Probability of error for the features with SVM trained
0.2
for only one value of SNR is comparable to the case where
0.15 SVMis trainedforeveryvalueofSNR. Moreover,noapriori
0.1 informationis requiredabout carrier amplitude, carrier phase,
carrier offset, symbol rate, pulse shape and initial symbol
0.05
phase (timing offset).
0 Simulation results showed that for quite low oversampling
−5 0 5 10 15 20
SNR [dB] the proposed classifier performs well while wavelet-based
classifier simply fails to classify. In order to have better
Fig. 2. Probability of misclassification (Pe). “TO h=x” represents the error performance, the number of samples per symbol and/or
proposedfeaturesforh=xwhereSVMistrainedonlyonceat10dB,“(WAV) symbols have to be increased. Furthermore, the number of
h=x”representsthe(wavelet-basedfeature)proposedfeaturesforh=xwhere
samples needed for reliable classification by the proposed
SVMistrainedforeveryvalue ofSNR.
features is less than that of wavelet feature.
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