MULTITIME-FREQUENCY CLASSIFIERS FOR NOISY MODULATIONS M. COLAS, G. GELLE, G. DELAUNAY Laboratoire d’Automatique et de Microélectronique (LAM) Université de Reims Champagne-Ardenne, BP 1039 51687 REIMS CEDEX 2 ; France e-mail : guillaume.gelle@univ-reims.fr Abstract : This communication presents two new classification algorithms based on Time Varying Higher Order Statistic (TVHOS). These two algorithms benefit from the advantageous localization properties provided by TVHOS4 for instantaneous frequency laws disrupted by a multiplicative noise. The classification scheme used is a bank of several normalized TVHOS4 correlators. Simulations illustrate successful performance of the classification algorithms for different situations and especially if the SNR is higher than –6 dB. I. Introduction Time Frequency and Higher Order Spectra have been intensively studied during these last few years. Recently, Time Varying Higher Order Spectra (TVHOS) are defined [1],[2] and permit to analyze non-linear time varying signals. In this paper, we present two new classification algorithms based on TVHOS applied to frequency modulations disrupted by multiplicative and additive noise. II. Time Varying Higher Order Spectra Many definitions of TVHOS can be found in the literature, they differ in particular in the lag separation between the time or frequency terms used for product. They can also differ in the number of conjugated terms and with the used space of representation as well : time- multifrequency space or multitime-frequency space. The user’s aim will lead him to decide upon the type of representation whether he chooses to set out the non linear phenomena or to preserve the time-frequency accuracy; for example in modulation cases. To reduce the computational cost, it is customary to consider only a slice of TVHOS. Sliced TVHOS (STVHOS) were first introduced by Fonollosa and Nikias in [3] and were defined as particular slices of the Wigner-Multispectrum. In practice, the principal slice of the Wigner-Trispectrum is the one most used for signal analysis because it is a real representation that contains all the autoterms of the signal. A computationally efficient implementation is given in the frequency domain by : 1 2 2 2 1 1 () 1 4 (, ) ( ). *( ) 4 4 j n xn SWD n X X e d πγ γ γ γ γ γ γ - = + - ∫ (1) Simultaneously, Stankovic [4] proposed a multitime- frequency definition of Wigner Higher Order Distributions as for the fourth order : ( ) 8 4 (, ) *( 1 2 3 ). ( 3 ) .(2 ). *( 1 ) xn k j k MTWD x n n n k xn k xn k x n ke πγ γ - = + + + - - - + ∑ n For computational purpose, evaluation of the MTWD4 can be done by considering only the principal temporal slice given by 3 2 1 n n n n = = - = . Hence, we obtain the L-Wigner Ville distribution : 2 2 8 ( ) 4 (, ) ( ). * ( ). j k xn k LW n x n k x n ke πγ γ - = + - ∑ (2) which is a dual formulation of (1). Due to its good localization properties, even for non-linear frequency modulation, LW4 has been extensively studied by Boashash in [5] for deterministic time varying signal processing. When dealing with random non stationary signals, Boashash defined the Moment-and the Cumulant Wigner-Trispectrum (MWD4 and CWD4 ) as : 8 () 4 8 () 4 4 (, ) (,) 4 (, ) (,) j k xn k j k xn k MWVD n m nke CWVD n c nke πγ πγ γ γ - - = = ∑ ∑ (3a & b) with { } ) ( * ). ( ) , ( 2 2 4 k n x k n x E k n m - + = ( E denotes the statistical expectation ) and { } ) ( * ). ( ) , ( 2 2 4 k n x k n x Cum k n c - + = (Cum represents the cumulant operator). [5] shows that this 2 formulations are helpful in multiplicative noise signals for instantaneous frequency law estimations. Signal analysis with TVHOS4 Let us consider the following model of signal : 1 1 () ( ). () m S n b nxn = (4) where ( ( ) ) 1 () j n xn e φ Φ + =