estimation of parameters of a polynomial phase model ... - IEEE Xplore

ESTIMATION OF PARAMETERS OF A POLYNOMIAL PHASE MODEL USING THE WARPED COMPLEX TIME DISTRIBUTIONS Cornel Ioana*, Srdjan Stankovic**, André Quinquis*

* : ENSIETA, Laboratory E3I2, EA 3876, 2 rue François Verny, Brest - FRANCE E-mail : [email protected], [email protected] ** : University of Montenegro, Podgorica – MONTENEGRO E-mail : [email protected]

ABSTRACT Polynomial modelling of the phases of non-stationary signals has received recently a great deal of attention. The main future of a such model is its capability to characterize accurately a large class of time-frequency non-linearities. The estimation of the polynomial model approximating the phase of a signal is typically based on high-order ambiguity function. Since a polynomial approximation of the phase is involved, two parameters have to be previously estimated. The first one is the most appropriate order of the polynomial model. The second parameter deals with time origin within the polynomial modelling at the “optimal” order remains valid. In this paper, we propose a method to estimate these parameters which define the behaviour of polynomial order. This method is based on the joint used of warping operators and complex time argument concept.

1. INTRODUCTION Signals encountered in engineering applications, such as communications, radar or sonar often involve amplitude (AM) and/or frequency modulation (FM). It is practically observed that the phase function of a large class of AM-FM processes can be modeled by a polynomial function depends of t. Consequently, these processes are called Polynomial-Phase Signals (PPSs) and are defined as follows :  K  s ( t ) = A exp jφ ( t ) = A exp  j ∑ ak t k   k =0 

(1)

where {ak} are the polynomial coefficients and K is the order of polynomial approximation of the phase of s. The estimation of polynomial coefficients is based on the high-order instantaneous-moment (HIM) defined as :

2. PARAMETRIZATION OF A PHASE POLYNOMIAL MODEL

HIMK s( t) ;τK-1 = HIMK−1 s( t +τK−1) ;τK-2 HIMK*−1 s( t −τK−1) ;τK-2 (2)

where τ i = (τ 1 , τ 2 ,..., τ i ) is the lag set. The main property of HIM is that, the Kth order HIM is reduced to a harmonic with amplitude A2

~ and phase , frequency ω

k −2

~

φ , when a significant component is present : k −1

HIM k  s ( t ) ; τ  = A2

(

exp j ω k ⋅ t + φk

)

(3)

a k (4). where ω~k = k!τ This property is exploited by taking the Fourier transform of HIM defining also the high-order ambiguity function (HAF). K −1

0-7803-9243-4/05/$20.00 ©2005 IEEE

Based on these results, Porat [1] has proposed an algorithm which estimates sequentially the coefficients {ak}. At each step, using a spectral analysis method, we estimate the spectral peak and, we compute an estimation value ( aˆ k ) of ak. Signals rising from real life applications have often multiple components and their parameter estimation poses a great challenge. Consequently, some techniques have been proposed, in order to reduce the influence of these cross-terms [1], [2]. The common point behind all these techniques is the initial assumptions concerning the behaviour of the polynomial model. More precisely, the highest polynomial order, K, must to be defined firstly. Furthermore, some additional parameters, such as the number of components, the time origin, etc, can be provided. Therefore, we consider that these items define the parameters of the polynomial model. The aim of this paper is to propose, in a realistic context (multi-component noised signals), a method for the estimation of these parameters whose definition is given in section 2. The principle is to project the signal in several subspaces, via warping operator techniques, and to estimate in each one the best mach between warped data and the assumed model. This estimation is done by the complex time distribution (CTD). Both items – warping operators and CTD are briefly described in section 3. The estimation of the phase model is given in section 4. Some results depicted for real data are presented in section 5. “Conclusions” (section 6) will highlight the significance of the results.

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In real cases, the polynomial phase model given in (1) is often inappropriate to describe the complexness of the signals. The first element which has to be taken into account in various applications (such as multipath effect characterization) is the multi-component behaviour of the received signal. Secondly, the choice of highest-order K could be a very peculiar problem especially in the case of strong nonlinear time-frequency contents. Namely, if a higher polynomial order K is required, the polynomial coefficients will be very small and instable against estimation errors or noise. Such situation is presented in the next figure. The phase approximation of the considered signal (“data 1” - solid line) requires eight order

polynomial function (dashed line). Polynomial coefficients are very small as indicated in figure 1. On the other hand, as depicted in [3], the estimation for high orders (superior to 5), the problem of error propagation affects drastically the performances of estimation. The method proposed in [3] could be a possible solution but its application in a multi-component context is reduced.

estimated polynomials and the phases of the signal component.

Fig. 2. Benefit of local polynomial modelling In the next section, we propose a method for the estimation of the polynomial phase model parameters.

3. WARPED COMPLEX TIME DISTRIBUTIONS Fig. 1. Polynomial modelling with a high order An alternative consists in expressing a given non-linear phase law by a serie of lower order polynomial phase models. Each model will be defined by two items : its order K and the temporal interval (origin and length) where the Kth-order model is valid. A signal composed by N components, whose phases are expressed as a polynomial of order Ki, can be written generally as : N N  K  (5) y ( t ) = ∑ Ai Ω ( t − τ i ) exp jφi ( t ) = ∑ Ai Ω ( t − τ i ) exp  j ∑ aki t k  D D i

i =1

i



i

i =1

k =0



th

where {aki} is the set of k -order polynomial coefficients of component i (i=1,..,N), Ki - is the order of polynomial phase modelling of component and Ω(t − τ i ) is the box Di

function of duration Di and origin τi defined as 1, t ∈ [τ i ,τ i + Di ] Ω(t − τ i ) =  Di otherwise 0,

(6)

The box function parameters and the order Ki constitute the “parametrization” of the polynomial model approximating the component i. Therefore, the polynomial phase model My of the signal y is parametrized as : (7) M y = {K i , (Di ,τ i )} For the signal whose phase is modelled in figure 1, (5) becomes y ( t ) ≈ exp  j 2π ( 0.04 ⋅ t + .51 ⋅10 −3 ⋅ t 2 + 26 ⋅10 −4 ⋅ t 3 )  ⋅ Ω ( t ) + 256

(8)

+ exp  j 2π (.21 ⋅ t + .43 ⋅10 ⋅ t + 65 ⋅10 ⋅ t )  ⋅ Ω ( t − 256 ) 256 −3

2

−4

3

This expression states that the considered signal can be expressed as the sum of two third order polynomials phase components whose length is about 256. Figure 2 illustrates both estimated instantaneous frequency laws (IFLs). We remark that, in terms of estimation accurateness, it is more advantageous to model the considered signal as the sum of two PPS than as a single one. Therefore, the appropriate setting of the polynomial model leads to the improvement of the match between the

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As stated by the model (5), a signal having a complex time-frequency behavior can be expressed as the sum of independent low order PPSs. Therefore, in spite of the simplified description way, the time-frequency structures composing the signal remain non-linear. Their processing can be done by the warping-based techniques [4]. For the signal s(t), it is defined as an operator U on l2(ℜ), i.e.:

( Us )( x ) =

w '( x)

1/ 2

s  w ( x ) 

(9)

where w is a smooth, one-to-one function [4], called warping function. Generally, this function is chosen to ensure the “linearization” of the signal’s time-frequency behavior. Therefore, for a signal expressed as s ( t ) = e j 2π cψ ( t ) , the associated warping function, w, is defined as the inverse of ψ (t), where ψ(t) is the frequency modulation law and c is the modulation rate. Thus, the linearization should enable that the time-frequency representation (TFR) of the warped signal produces a constant instantaneous frequency. Distributions from the Cohen’s class (CTFR) are usually used for these TFRs. Nevertheless, in the case of multi-component signals, the linearization of the time-frequency behaviour of the warped signal will be embedded by some artefacts corresponding to application of the warping operator on the signal components having a different behaviour than the one associated to the warping operator. An alternative is to use the complex time argument concept [5] instead of CTFRs. The main property of the complex time distribution (CTD) consists in the capability to attenuate high-order terms of the polynomial decomposition of the IFL. As proved in [5], in the case of the CTD, the spread function (i.e. the function expressing analytically the spreading of time-frequency energy, caused by the non-linear artefacts terms, around the true IFL) has a fifth order dominant term (for comparison, the spectrogram and the WVD have a second and a third order dominant term, respectively), which corresponds to a drastic reduction of the higher terms for a non-linear phase law. This property is illustrated in the following example where we consider the warped version of the signal :

s (t ) = e

(

j 2π t 0.2 + 0.75 t 0.7

)

(10) We use the power warping operator [4] which consists in changing the time axis according to t0.7. In figure 3 the Wigner-Ville distribution (WVD) of the warped signal is plotted. Since the term t0.7 has been compensated through the warping operator, we remark that the IFL is more “linear” than the original version. Nevertheless, since the term t0.3/0.7 occurs, the linearization of the time-frequency behaviour is not achieved.

representation is concentrated around the parameter of the IFL (tk in the case of polynomial model) corresponding to the warping operator. If the signal does not contain such IFL, the spread function Q dominates the CTD representation. In order to evaluate the concentration of the CTD warped signal according to (12), the frequency marginal function can be used :

Λ U K (ω ) = t i

∞

∫ CTDU ( t , ω ) dt t Ki

−∞

x

(13)

CTDU K x ∈WCTDx ; Ki = 1,.., P t i

Fig. 3. CTD vs WVD of a warped signal We can observe also that CTD provides a better linearization reducing considerably the level of artefacts. For this reason, the combination of the CTD with a family of warping operators (Warped Complex Time Distribution-WCTD) produces an efficient characterization of a signal whose phase components correspond to the operators of this family [6]. Analytically, the WCTDs are defined as the set of the CTDs of the analysed signal x whose time-frequency content was deformed according to several warping operators (Ua) :

{

WCTDx ( t , ω ) = CTDU CTDU

ax

ax

( t , ω )}

(11)

( t , ω ) = 2πδ (ω − a ) ∗ω FT {e j 2π Q(t ) }

spread function which affects the IFL visibility associated to a. In the next section, the WCTD technique will be adapted for the parameter polynomial order estimation purposes.

4. POLYNOMIAL MODEL ESTIMATION USING THE WCTD In order to estimate the parameters of a polynomial model, the family of warping operator is chosen according to the polynomial term tKi for Ki spanning the interval from 1 to a maximal value of interest P. The CTD of the corresponding warping operator, applied on the signal (5), is expressed as : N j 2 π Q (t )  C T D U s = ∑ Ω ( t − τ i )  2 πδ (ω − a i , K ) ∗ ω F T {e } (12)  D i =1

i

i

th

As this relation states, the term associated to K i order

δ (ω − ai , K

i

)

t Ki

functions Λ U defined in (13). K t i

Furthermore, as stated in (12), the CTD, associated to a given polynomial order, contains several distinct shorttime tones ai,Ki whose duration and origin are related to the parameters ( Di ,τ i ) of the polynomial model (5). Their estimation could be done thanks to the WCTD timemarginal

ΨU K (t ) = t i

where a is the parameter of the frequency modulation law associated to the warping operator Ua, FT is the Fourier transform, ∗ω is the convolution operator and Q(t) is the

tKi

In the case of the polynomial model, the most concentrated CTD will be obtained for the warping operator associated to the most appropriate polynomial order. More precisely, the estimation of this order is provided by looking for the most concentrated function Λ U , i.e., the most energetic peak of the set of

is convolved with artefacts generated by

warping the other polynomial components. These artefacts are mathematically represented by Q. As illustrated in the figure 3, the CTD attenuates this function and the 637

∞

∫ CTDU ( t , ω ) dω t Ki

−∞

x

(14)

by evaluating the argument of the local maximum of its gradient :

τˆi = arg max  d Ψ U / dt  t



t Ki



(15)

Dˆ i = τˆi +1 − τˆi The existence of two different frequency tones in the neighbour of a time instance τˆi is introduced by the maximum of the gradient of Ψ U . The interval between K t i

two consecutive maxima leads to the duration of the Di. Therefore, a complete characterisation of the polynomial model can be done. Furthermore, an estimation procedure of polynomial coefficients can be properly set up.

5. RESULTS In this section we illustrate the WCTD-based method for estimation of the polynomial model in the case of a real signal. Considering the problem of blind estimation of the impulse response (IR) of a real underwater multi-path channel (oceanic passive tomography context [7]), this signal is composed by the sum of several attenuated and delayed versions of an emission characterized by a nonlinear time-frequency behaviour. The spectrogram of the received signal and the IR of the channel are illustrated in the figure 4.

This satisfactory result is mainly explained due to the estimation method performances, which are conditioned by the model parameterization provided by WCTD.

Fig. 4. Spectrogram of the signal (a) issued from a channel whose IR is defined in (b)

The evaluation of the IR, without any knowledge about the transmitted signal (passive tomography context), could be done by modelling the received signals as multicomponent PPS. Application of such processing requires firstly the estimation of the polynomial model parameters. Results provided by the WCTD method are depicted in the figure 5.

Fig. 6. Estimated IR

6. CONCLUSION Since the polynomial modelling of the signal phases constitutes a useful way for characterisation of non-linear TF behaviour, we address in this paper the problem of parameterization of a polynomial model. Namely, using a method based on the joint use of warping techniques and CTD, we propose an estimation procedure for the polynomial order and its time validity intervals. The benefits of this model have been devised through an example. Off course, we don’t claim that the model (5) covers all the situations meet in real applications, but it could be successfully assumed in several cases. An application considered in this paper, is the non-linear time frequency signals received in a multi-path configuration. Results proved the capability of this method to provide an efficient characterization of the polynomial model in spite of multi-path item. Acknowledgments. This work was supported by the French Military Center of Oceanography under the research contract CA/2003/06/CMO.

Fig. 5. Estimation of parameters of polynomial order

REFERENCES

Figure 5.a. illustrates the WCTDs frequency marginals provided for polynomial orders between 2 and 5 (relation (13)). Obviously, the more energetic peak is obtained for k=3 and it constitutes the most appropriate modelling order for the processed signal. Furthermore, the WCTD of order 3 (figure 5.b.) shows the existence of two distinct stationary time-frequency groups separated at the instant 5.5 seconds. This instant can be computed, via the relation (15), from the WCTD time marginal (figure 5.b.). According to these parameters, we conclude that the processed signal can be modelled as a sum of two groups of PPSs having the same order 3 and the durations 5.5s and 2.5s, respectively. This model is similar to the one considered in figure 2. Assuming a signal with a complex time-frequency behaviour, WCTD leads to the parameters of a simplified model but is more attractive for further processing. The correctness of this model is justified by the estimated IR trough a polynomial modelling method proposed in [7]. As illustrated in the figure 6, the estimated IR is close to the real one depicted in figure 4.a.

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[1] B. Porat, Digital Processing of Random Signals, Pretince Hall, 1993. [2] S. Barbarossa, A. Scaglione, G.B. Giannakis, “Product HighOrder Ambiguity Function for Multicomonent Polynomial-Phase Signal Modeling”, IEEE Transactions on Signal Processing, vol. 46, No. 3, 1998. [3] A. Quinquis, C. Ioana, E. Radoi, “Polynomial Phase Signal Modeling Using Warping-Based Order Reduction”, ICASSP 2004, Montreal, Canada. [4] A.Papandreou-Suppappola, ed., “Applications in timefrequency signal processing”, CRC Press, Boca Raton, 2003. [5] S. Stankovic, L. Stankovic, “Introducing time-frequency distributions with a complex time arguments”, Electronic Letters, vol. 32, No. 14, pp. 1265-1267, July 1996. [6] C. Ioana, S. Stankovic, A. Quinquis, LJ. Stankovic, “Modelling of signal’s time-frequency content using Warped Complex-Time Distributions”, ICASSP 2005, Philadelphia, USA. [7] C. Ioana, C. Gervaise, A. Quinquis, “Blind deconvolution of underwater channel using transitory signal processing”, Proceedings of OCEANS 2004, pp 1033-1036, Kobe, Japan.