HOS Based Distin tive Features for Preliminary Signal Classi ation Ma iej Pdzisz and Ali Mansour É ole Nationale Supérieure des Ingénieurs des Études et Te hniques d'Armement (ENSIETA), Laboratoire Extra tion et Exploitation de l'Information 3 2 en Environnements In ertains (E I ), Brest, Fran e,
pedzismaensieta.fr, mansourensieta.fr
Abstra t.
We onsider the problem of preliminary lassi ation of dig-
itally modulated signals. The goal is to simplify further signal analysis (syn hronization, signal separation, modulation identi ation and parameters estimation) by making initial separation among the most known
lasses of signals. Proposed methodology is mainly based on Higher Order Statisti s (HOS) of the distributions of instantaneous amplitude and frequen y. The experimental results emphasize the performan e of the proposed set of features.
1 Introdu tion In Communi ation Intelligen e (COMINT), knowledge of signal's frequen y stru ture is essential to re ognize underlying modulation type and measure its parameters. Up to now, all frequen y syn hronization algorithms onsider only one signal, they need a big number of symbols and a long time to onverge. Thus, making a preliminary signal lassi ation based on frequen y invariant features, will mu h simplify further pro essing, allowing appli ations of signal-spe i syn hronization, sour e separation and modulation lassi ation te hniques. In [1℄, authors presented empiri al results in Blind Sour e Separation (BSS) using over omplete Independent Component Analysis (ICA) representations. They demonstrated delity of their algorithm in the ase of 2 mixtures of 3 spee h signals. Separation of 2 audio sour es from a single sensor is the subje t
overed in [2℄. Proposed method generalizes the Wiener ltering with Gaussian Mixture distributions and Hidden Markov Models. A time-frequen y ltering based on the Pseudo Wigner-Ville distribution is onsidered in [3℄. Performan e of the presented algorithm was validated using a mixture of 2 voi e re ordings. In [4℄, sparse fa torization approa h with K-means lustering algorithm applied to BSS problem is dis ussed. Provided results reveal the performan e of the algorithm in ase of 10 fa e images (6 mixtures), as well as 8 spee h signals (5 mixtures). Authors of [5℄, derive algebrai means for ICA in the ase of undetermined mixtures. Their results are based on the stru ture of the fourth-order
umulant tensor. Sixth-order statisti s and the virtual array on ept are addressed in [6℄. It was shown that their algorithm an be used to in rease the C.G. Puntonet and A. Perieto (Eds.): ICA 2004, LNCS 3195, pp. 1158-1164, 2004
Springer-Verlag
Berlin Heidelberg 2004
HOS Based Distin tive Features for Preliminary Signal Classi ation
1159
ee tive aperture of an antenna array, and so to identify the mixture of more sour es than sensors. The ase of binary sour e separation is overed in [7℄ and [8℄. Their algorithm uses the stru ture of the probability distributions of the observed data. Simulations showed that the method an su
essfully separate at least up to 10 binary sour es at dierent noise levels. On the other hand, modulation re ognition algorithms ([9℄, [10℄, [11℄, [12℄) deal with the ases where some a priori information is available ( arrier frequen y, symbol timing, ...) and there is only one signal in additive noise. In this ontribution, we try to ll the gap between syn hronization & modulation re ognition methods, and sour e (signal) separation algorithms based on one observation (undetermined problem). Using the proposed set of features, we are able to distinguish among the most ommon known signal types, and so, hoose the appropriate methodology for further signal pro essing.
2 Signal Models 2.1 Mono-Component Signal Let's assume working in the onditions where signal's arrier frequen y is not known. The re eived omplex baseband signal (after imperfe t demodulation)
an be expressed as a sum of two un orrelated omponents:
s(t) = Ac (t)ej(ωr t+Θr ) + n(t)
(1)
where Ac (t) is a signal omplex envelope, ωr is a residual frequen y, Θr is a phase of the residual frequen y, and n(t) orresponds to a zero-mean, additive white gaussian omplex noise. Using the on ept of the omplex envelope, we an express any linearly modulated signal as:
Ac (t) = A
X
dk h(t − kT − τ ),
k ∈ {1, 2, . . . , K}
(2)
k
where A is a onstant amplitude, dk des ribe signal onstellation, h(t) is a pulse shaping fun tion, T is a symbol duration, τ is an out-of-syn hronization error (due to imperfe t demodulation), and K is a number of available symbols. For the most known M-ary linear modulations (MASK M-ary Amplitude Shift Keying, MQAM M-ary Quadrature Amplitude Modulation, MPSK M-ary Phase Shift Keying), we have:
dMASK = ak , k dMQAM = ak + k MPSK dk = ejϕk ,
ak ∈ {±(2m − 1) : m = 1, 2, . . . , M/2}
(3)
jbk , ak , bk ∈ {±(2m − 1) : m = 1, 2, . . . , log2 (M ) − 2} (4) ϕk ∈ { 2π M (m − 1) : m = 1, 2, . . . , M } .
(5)
In the nonlinear ase (MFSK M-ary Frequen y Shift Keying), we an write:
Ac (t) = Aej
P
k
dk ∆ω (t−kT −τ )h(t−kT −τ )
,
k ∈ {1, 2, . . . , K}
(6)
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Ma iej P edzisz and Ali Mansour
where ∆ω is a frequen y deviation, and dk an be expressed as:
dMFSK ∈ {±(2m − 1) : m = 1, 2, . . . , M/2} . k
(7)
It is assumed that variables ak , bk and ϕk in equations (3), (4) and (5), as well as dk in (7) are independent and identi ally distributed (i.i.d. pro esses). It is assumed also that all modulation states are equiprobable (whi h is always a
omplished when sour e oding is applied) and the pulse shaping fun tion h(t) is re tangular.
2.2 Multi-Component Signal Taking into onsideration the mono- omponent model of a linear modulation ((1) and (2)), we an write a general formula for a multi- omponent signal as:
S(t) =
L X
=
L X
Aci (t)ej(ωri t+Θri ) + ni (t)
i=1
i=1
(8)
Ai
X
dki hi (t − kTi − τi )ej(ωri t+Θri ) + n(t)
k
where L is a number of mono- omponent signals and n(t) is a term whi h absorbed all noise ontributions ni (t). In COMINT appli ations, it is often su ient to onsider: there are two signals in the mixture (L = 2), and applied modulation types are MPSK. Additionally, we assume that signal amplitudes are identi al (A1 = A2 )1 . We are not
onsidering prior knowledge about:
3
residual frequen ies and phases (ωri and Θri ); symbol durations (Ti ); syn hronization errors (τi ).
Distin tive Features
3.1 Preliminary Results Based on the signal models (1) and (8), we an rewrite the re eived signal as:
sr (t) = p(t) + jq(t) = Ai (t)ejφi (t)
(9)
where p(t) and q(t) are in-phase and quadrature omponents, Ai (t) is an instantaneous amplitude and φi (t) is an instantaneous phase. Then, we an de ne:
Ai (t) = |sr (t)|, 1
General ase
φi (t) = arg{sr (t)},
A1 6= A2
ωi (t) =
dφi (t) dt
will be addressed elsewhere.
=
dq(t) dp(t) p(t) dt −q(t) dt 2 2 p (t)+q (t)
(10)
HOS Based Distin tive Features for Preliminary Signal Classi ation
1161
where ωi (t) is an instantaneous frequen y. In general, ωi (t) is de ned using the
on ept of the analyti signal [13℄. It is well known that the probability density fun tion (PDF) of Ai of any MPSK/MFSK signal an be expressed in terms of its onstant amplitude A (Eq. 2 (2)) and noise varian e σn (Eq. (1)) by means of Ri e distribution [14℄: 2
fAi (Ai ; A, σn2 )
2
Ai − Ai +A = 2 e 2σn2 I0 σn
Ai A σn2
,
Ai > 0
(11)
where I0 (x) is the modi ed Bessel fun tion of order 0. If A = 0 (NOISE), then PDF of Ai be omes Rayleigh. For the MQAM lass 2 ) over of signals, we an write the orresponding PDF as a mean of fAi (Ai ; Al , σn all distin tive amplitudes Al . The se ond distribution whi h an be onsidered as distin tive in signal lassi ation is the PDF of ωi [15℄. For a single- arrier modulation (MPSK, MQAM), we have:
A2 3 , 1; 2 2σn2 vi
2
A − 32 − 2σ 2
fωi (ωi ; A, σn2 ) = ϑ−1 vi
e
n
1 F1
R +∞
(12)
R +∞
where vi = 1 + ωi2 /ϑ2 , ϑ2 = −∞ ωi2 γ(ω) dω/ −∞ γ(ω) dω , γ(ω) is a power spe tral density (PSD) of noise, and 1 F1 (α, β; x) is a on uent hypergeometri fun tion de ned as: 1 F1 (α,
β; x) =
+∞ X Γ (α + k)Γ (β)xk k=0
Γ (α)Γ (β + k)k !
β 6= 0, −1, −2, . . .
,
(13)
It is obvious that in the multi- arrier ase (MFSK), the PDF of ωi an be expressed as a mean over all distin tive ( arrier) frequen ies. Finally, when A ≫ σn , we an approximate both distributions by the Gaussians [13℄, [15℄, [16℄: √n ) fωi (ωi ; A, σn2 ) ≈ N (ωi ; 0, 2Bσ 3A
fAi (Ai ; A, σn2 ) ≈ N (Ai ; A, σn2 ), where N (x; µ, σ 2 ) ,
√1 σ 2π
(14)
i h 2 , and B is a noise ee tive bandwidth. exp − (x−µ) 2 2σ
3.2 Features Extra tion The main obje tive in preliminary signal lassi ation is to nd a set of hara teristi s whi h allows distin tion among dierent lasses of signals. Based on distributions of Ai and ωi , we an extra t normalized umulants [17℄ of order 3 γ3 (skewness) and 4 γ4 (kurtosis) as:
γ3 =
κ3 3/2 κ2
,
γ4 =
κ4 κ22
(15)
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Ma iej P edzisz and Ali Mansour
where umulants κr and orresponding moments mr are de ned by:
κ2 = m2 − m21 κ3 = m3 − 3m2 m1 + 2m31 κ4 = m4 − 4m3 m1 − mr =
Z
+∞
3m22
(16) (17)
+
12m2 m21
−
6m41
r
(18) (19)
x f (x) dx .
−∞
Other sets of hara teristi s an be obtained by using Renyi's quadrati entropy [18℄:
H2 = − log
Z
+∞
2
f (x) dx −∞
(20)
and by solving a polynomial regression on the logarithm of a PDF:
log(f (x)) ≈
X
ak xk .
(21)
k
3.3 Features Sele tion & Dimensionality Redu tion It is obvious that limiting the number of features will make learning and testing faster and demanding less memory. Aside from this, feature spa e of a lower dimension may enable more a
urate lassi ers for a nite learning set. Based on the hara teristi s presented in the previous se tion, experiments have been ondu ted to hoose the most dis riminative set of features:
A A A features based on Ai : γ3A , γ4A , H2A , aA 3 , a2 , a1 , a0 (3-rd degree polynomial is su ient to des ribe asymmetry and atness of onsidered distributions); ω ω features based on ωi : γ4ω , H2ω , aω 4 , a2 , a0 (PDF of ωi is symmetri al about the mean, so all the features based on asymmetry were eliminated).
On e they have been sele ted, one an apply the Linear Dis riminant Analysis to verify the importan e of hosen features. Using the Fisher's riterion [19℄:
JF = tr{T} = tr{S−1 w Sb }
(22)
where Sw is the within- lass ovarian e matrix (the sum of ovarian e matri es omputed for ea h lass separately), and Sb is the between- lass ovarian e matrix (the ovarian e matrix of lass means), we found:
all sele ted features are of equal importan e among dierents ombinations of features, the whole set is the most dis riminative; features from Ai are best to separate between lasses of signals with symmetri Ai PDF (MPSK, MFSK) and asymmetri (NOISE, MQAM and MIXTURE); features from ωi are best to separate between lasses of signals with unimodal ωi PDF (MPSK, MQAM and MIXTURE) and multimodal (MFSK).
HOS Based Distin tive Features for Preliminary Signal Classi ation
1163
It should be noted, that using eigenve tors of matrix T, it is possible to redu e dimensionality of the feature ve tor A A A ω ω ω ω ω T x = [γ3A , γ4A , H2A , aA 3 , a2 , a1 , a0 , γ4 , H2 , a4 , a2 , a0 ]
(23)
by means of linear transformation: (24)
y = Wx
where eigenve tors orresponding to largest eigenvalues of T form the rows of the transformation matrix W.
4
Simulations
To evaluate the performan e of the proposed set of features, extensive simulations were ondu ted on the signals: NOISE, MPSK (2, 4 and 8), MFSK (2 and 4), MQAM (16 and 32) and MIXTURE (2xBPSK, 2xQPSK and BPSK & QPSK). All signals were omposed of 512 samples, 5 samples per symbol, 1000 dierent realizations. Signal to Noise Ratio (SNR) was varying from 0 dB up to 30 dB. The residual frequen ies ωri , the orresponding phases Θri , as well as the symbol timings Ti , were hosen randomly a
ording to Nyquist sampling theorem. Corresponding results (SNR = 5 dB) are shown in Fig. 1.
SNR = 5 dB 5
MFSK
MPSK 3
1 y2
MQAM
−1
−3 MIXTURE
−5 −5
−3
NOISE
−1
1
3
5
y1
Fig. 1. Signals in a 2D spa e after dimensionality redu tion (SNR = 5 dB).
1164
Ma iej P edzisz and Ali Mansour
5 Con lusion It is evident that sele ted set of features is very e ient even for low SNR. Perfe t lassi ation an be obtained for the lasses NOISE, MPSK and MFSK for SNR > 5 dB, however distin tion between MQAM and MIXTURE is far from being su ient enough. Although lassi ation in a 2D spa e was used for visualization purposes, one should not limit himself during onstru ting a nal lassi er. Adding another set of hara teristi s (based for example on Time-Frequen y Distributions (TFD)), may be more attra tive in more than 2 dimensions. Also, making lassi er hierar hi al or using some nonlinear mappings (MMI [20℄, NPCA [21℄), may in rease separability of the lasses. These topi s will be overed in a future work.
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