Spectral Analysis by Stationary Time Series Modeling

Transcription

1 Chater 6 Sectral Analysis by Stationary Time Series Modeling Choosing a arametric model among all the existing models is by itself a difficult roblem. Generally, this is a riori information about the signal and it makes it ossible to select a given model. For that, it is necessary to know the various existing models, their roerties, the estimation methods for their arameters and the various ossible imlementations. In this chater, we resent the various arametric models of stationary rocesses. Chater 4 brought an introduction to this chater by resenting the models of the time series: the ARMA and Prony models have thus already been introduced. The subject of this chater is to go through these models by tackling their arameter estimation roblem as well as their imlementation form. 6.. Parametric models The stationary arametric models discussed in this chater can be regroued into two categories: ARMA models, Prony models. Chater written by Corinne MAILHES and Francis CASTANIÉ.

2 5 Sectral Analysis The geometrical methods, based on the decomosition of a signal sub-sace and of a noise sub-sace (MUSIC, ESPRIT, etc.), secific of sinusoidal signals, are resented in Chater 8. These two tyes of models were resented in Chater 4, but in order to make the reading of this chater easier, we will recall the fundamental equations of various models. As most of the rocesses can be well aroximated by a linear rational model, the ARMA model (, q) is an interesting general model. It is described by a finite order linear recursion with constant coefficients: = n ( ) + n ( ) [6.] x k a x k n b u k n 0 in which u(k) reresents a stationary white noise with null mean. q The coefficients a n corresond to the AR art of the model, while the coefficients b n determine the MA art. A articular case of the ARMA models is the AR model of order, which is written: n x( k) = a x( k n) + u( k) [6.] Autoregressive modeling (AR) is certainly the most oular analysis in arametric sectral analysis. The interest of this method lies in the ossibility of obtaining very recise estimations of the AR arameters even for a small number of samles. In order to have a more hysical descrition of the signal, in terms of frequency, amlitude, hase and daming, the Prony model can also be envisaged. This model consists of describing the signal as a sum of comlex exonentials. This model is thus a riori deterministic, contrary to the ARMA and AR models. Thus, it makes it ossible to model deterministic signals or moments (autocorrelation functions, for examle) by: x k m= k m m = B z

3 Sectral Analysis by Stationary Time Series Modeling Estimation of model arameters In the roblem of the estimation of various model arameters, we first take interest in the estimation of AR arameters because this roblem is also found again in the ARMA and Prony modelings. We will then tackle the roblem of the estimation of the ARMA and Prony model secific arameters Estimation of AR arameters Let us suose that we wish to aroach an ordinary signal x(k), for the interval k = 0,..., N, by the model [6.]. The method, which is most currently used to calculate the autoregressive coefficients, consists of minimizing the Linear Prediction Error (LPE) in the mean square sense. By considering the recursion equation [6.], we can realize that we can build a linear redictor of the signal both in the forward and backward direction. We define the forward redictor ˆx ( k ) of x(k) by: xˆ ( k) = anx( k n) [6.3] We then define the error of forward linear rediction by: n e( k) = x( k) xˆ ( k) = x( k) + a x( k n) [6.4] and the error of backward linear rediction by: = ( ) + ( ) b k x k a x k n + n [6.5] AR a n n The arameters { } =, quadratic criterion: are estimated so that they minimize the following min σ e = ak N e( k) [6.6] k= and/or the equivalent backward criterion min { ak b } σ. If the signal is efficiently modeled an AR model of a given order, we can show [KAY 88] that the coefficients calculated this way are good estimators of the a of equation [6.] if the order of the chosen model is equal to. coefficients { } n

4 54 Sectral Analysis The solution of this least-squares roblem is exressed in the following way: aˆ H H aˆ = ( X X) X x with aˆ = [6.7] aˆ Deending on the chosen minimization window the matrices and the vectors X and x are defined in different ways. In the case where: ( ) ( 0) x x x X= X = x = x = x( N ) x( N ) x( N ) [6.8] this estimation method is imroerly called the covariance method because the H H matrix X X and the vector X x are estimations of covariances with one normalizing coefficient. Morf has given an order recursive algorithm making it ossible to calculate this solution without exlicit inversion of the matrix [KAY 88, MOR 77]. By adoting a least-squares solution, which minimizes the forward [6.6] and backward sum of the rediction errors, we choose: x* N x* N x* x* X= X = x( ) x( 0) x( N ) x( N ) [6.9] x* N x *0 x= x = x x( N ) This is the modified covariance method and it is generally more efficient than the covariance method. Sometimes it is also called the maximum entroy method

5 Sectral Analysis by Stationary Time Series Modeling 55 (MEM) [LAN 80] because it is a articular case when the noise is Gaussian. Burg develoed an order recursive algorithm making it ossible to obtain the reflection coefficients that minimize the sum of the forward and backward rediction errors, which makes it ossible to deduce the AR arameters [BUR 75] via the Levinson- Durbin recursion. The advantage of this algorithm is that the estimated oles are always inside or on the unit circle. Its main disadvantages are a sectral line slitting in the case of a signal made u of a noisy sinusoid with a strong signal-to-noise ratio (SNR) and sensitivity at the initial hase [SWI 80, WIL 93]. This sensitivity at the hase can be simly highlighted by considering the following case: = cos( π + φ) + x k fk u k () a aˆ x x 0 x ˆ = aˆ = x x x 3 [6.0] The sensitivity of the solution â is evaluated through the conditioning κ of the system [6.0] [GOL 89]: λ κ = [6.] λ max min or λ max and λ min are the eigenvalues of the following matrix: ( π f φ) cos( φ) ( π fn + φ) ( π f + φ) cos + cos 4 cos ( f ) ( f ) λmax = cos π + φ + cos φ cos 4π + φ λmin = cos( π f + φ) cos( φ) cos( 4π f + φ) [6.] The difference between these two eigenvalues imacts on the extent of the errors π in â. It is evident that when φ =, the eigenvalues are identical and the errors in a are minimal. When the estimation of the AR arameters is realized in the leastsquares sense, the sensitivity at the initial hases of the sinusoids diminishes but it can remain imortant if the number of samles is small. But the estimation of the AR arameters of an autoregressive model of order can also be done starting from:

6 56 Sectral Analysis xx x xx n ( m) = E x( k) x* ( k m) = a ( m n) + ( m) γ γ σ δ [6.3] which we call Yule-Walker equations, which were already mentioned in Chater 4. This result shows that the autocorrelation of x(k) satisfies the same recursion as the signal. A large number of estimation methods solve the Yule-Walker equations by m γ m : relacing the theoretic autocorrelation γ by an estimation xx xx xx xx xx * * ( ) * () ( 0) ˆ - Rx () ˆ γ 0 ˆ γ ˆ γ ˆ γ ˆ ˆ ˆ ˆ xx ˆ γxx γxx γxx γ a = ˆ γ ˆ ( 0) ˆxx xx γ γ xx rˆ x ˆxx [6.4] It is the autocorrelation method and R ˆ x is the (estimated) autocorrelation matrix. When ˆxx γ ( m) is the biased estimator of the correlation, the oles are always inside the unit circle, which is not the case with the non-biased estimator which gives, however, a better estimation. The Levinson-Durbin algorithm [LEV 47] rovides an order recursive solution of the system [6.4] in O( ) comutational burden. In order to reduce the noise influence and to obtain a better estimation, the system [6.4] can use a larger number of equations (>> ) and be solved in the leastsquares sense (LS) or in the total least-squares sense (TLS) [VAN 9]. These methods are called LSYW and TLSYW (see [DUC 98]): Rˆ x LSYW TLSYW ( ˆ H ˆ ) ˆ H x x x ˆx H ( σ min I ) aˆ = R R R r * ( ) H x x x ˆx aˆ = Rˆ Rˆ Rˆ r () ˆ γ 0 ˆ xx γxx ˆ γxx ˆ rx = ˆ γ ( 0 ) ˆ ( 0 ) ˆ ( 0) xx N γxx N γxx N [6.5] where σ min is the smaller singular value of the matrix ˆ x ˆx R r, I the identity matrix and N 0 the number of equations. The value of N 0 should be at most of the order of N when the non-biased estimator of the correlation is used so as not to make correlations with a strong variance intervene. The TLS solution is more efficient than the LS method because it minimizes the errors in r ˆx and in R ˆ x at the same time, but it resents a disadvantage which we

7 Sectral Analysis by Stationary Time Series Modeling 57 will encounter later. It is generally better to envisage a calculation of the LS and TLS solutions via the singular value decomosition (SVD) of the matrix R ˆ x, articularly when the system is not well conditioned, because the SVD roves a greater numeric stability. The LS and TLS solutions calculated by the SVD are: ( ˆ ˆ ) Rˆ Uˆ ˆ Vˆ ˆ ˆ ˆ H x = Σ Σ = diag σ,, σ σk σk+ ( ˆ ) ˆ H Σ ˆ ( Σˆ ) diag ( ˆ σ ) ˆ σ LSYW aˆ = Vˆ U r =,, x aˆ v + TLSYW = v + ( + ) [6.6] where V + is the eigenvector associated to the smallest eigenvalue of the matrix H ˆ ˆ ˆ x x x ˆx R r R r, i.e. the ( + ) th column vector of the matrix V ˆ + so that ˆ ˆ ˆ ˆ H x ˆx R r = U+ Σ + V+. There exists an extension of the Levinson-Durbin algorithm for solving LSYW, it is what we call the least-squares lattice [MAK 77, PRO 9]. The interest of this algorithm is that it is time and order recursive, which makes it ossible to imlement arameter estimation adative rocedures. We can very easily notice that these methods are biased because the correlation at 0 th lag makes the ower of the white noise σ u intervene. The aroached value of this bias for (6.4) is: ( ˆ σ I ) + aˆ = R r ˆ x u u x u aˆ a+ σ Rˆ r = a Rˆ a [6.7] u x uˆx u σu x u where x u is the noiseless signal. Certain methods [KAY 80, SAK 79] exloit this relation in order to try to eliminate this bias, but these require an estimation of σ u. The simlest solution for obtaining a non-biased estimation of the AR arameters consists of not making the zero lag correlation intervene in the estimation: ( + ) ˆ γxx ˆ γxx ˆ γxx ˆ a = ˆ γxx ( ) ˆ γxx ˆ γxx ( ) [6.8]

8 58 Sectral Analysis That is called the modified Yule-Walker method (MYW). When we solve this system in the classic or total least-squares sense, we call these methods LSMYW and TLSMYW [STO 9]. When the signal x(k) is really an AR(), these estimators are asymtotically unbiased ( N ). However, if x(k) is made u of a sum of sinusoids and white noise, the MYW estimators are also asymtotically unbiased [GIN 85]. The case of a colored noise slightly modifies the MYW estimators. The hyothesis that the noise u(k) is of the form of a moving average rocess (MA) of order q is the most common: q n 0 u( k) = b ε ( k n), [6.9] where ε(n) is a null mean Gaussian white noise. In this case, the correlation γ ( m ) of u(k) is null when m > q. Thus, an estimator of the AR arameters of an ARMA (, q) or of a sum of noisy sinusoids by a MA(q) is: xx ˆ γxx + q ˆ γxx q+ ˆ γxx + q+ ˆ a = ˆ γxx ( + q ) ˆ γxx ( + q) ˆ γxx ( + q) [6.0] Of course, identifying the noise structure in ractice is far from being evident and very often we satisfy ourselves with the whiteness hyothesis. These methods suose that the order is known. In ractice it is not necessarily simle to determine it (we will see the methods further) but it should not be underestimated in order not to forget to estimate all the signal oles. When it is overestimated, besides the signal oles, oles linked to the noise also aear; they are more or less damed but in any case they are generally more damed than the signal oles (if the SNR is not too weak). This makes it ossible to distinguish between the signal oles and the noise oles. The LSMYW method of an overestimated order is called high-order Yule-Walker method (HOYW). An order overestimation resents the advantage of imroving the signal oles estimation. The LSMYW method is relatively efficient in order to distinguish between the signal and the noise oles, on the contrary the TLSMYW method has the disadvantage of attracting the oles (including those of the noise) on the unit circle making the distinction very difficult. We will show this henomenon on one examle; we consider N = 00 samles of a signal made u of non-damed sinusoids of frequencies 0. and 0.3 and of a white noise such as SNR = 0 db. The oles are estimated by the LSMYW and TLSMYW methods with a number of equations N

9 Sectral Analysis by Stationary Time Series Modeling 59 and an order = 0. Figure 6. reresents the lot of the oles estimated in the comlex lane for a signal realization. We notice that the oles linked to the noise of the LSMYW method are clearly damed in relation to those obtained with TLSMYW which sometimes fall outside the unit circle. When the order is overestimated, the TLSMYW method should be avoided. However, for a correct order, the estimation by TLSMYW is better than that of LSMYW... Figure 6.. Influence of the order overestimation for the LSMYW and TLSMYW methods U to now, the LSMYW method is one of the most efficient of those that we have described [CHA 8, STO 89]. Stoica and Söderström [SÖD 93] have given the asymtotic erformances of this method in the case of a noisy exonential (white noise) of circular frequency ω = πf and amlitude A: Var ( ˆ ω) lim E ( ˆ ω ω) ( + ) 4 u σ = N = N A + N [6.] where reresents the order and N 0 the number of equations. This exression can be comared to the asymtotic Cramer-Rao bound of the circular frequency ω (see section 3.4.): 6σ u 3 CRB ω = [6.] A N Or we could think that the variance of HOYW [6.] can become smaller than the Cramer-Rao bound. Actually, this is not the case at all if it is only asymtotically that it can become very close to Cramer-Rao. [SÖD 9] comares this method to the Root-MUSIC and ESPRIT methods (see Chater 8) in terms of recision and

10 60 Sectral Analysis comutational burden. HOYW makes a good comromise between these two asects Estimation of ARMA arameters The estimation of ARMA arameters [KAY 88, MAR 87] is made in two stes: first, we estimate the AR arameters and then the MA arameters. This subotimal solution leads to a considerable reduction of the comutational comlexity with resect to the otimal solution that would consist of estimating the AR and MA arameters. The estimation can be efficiently erformed in two stes. Starting from equation [6.], we show that the autocorrelation function of an ARMA rocess itself follows a recursion of the AR tye but starting from the lag q + (see section 4.., equation [4.7]): γ γ m = a m n for m> q xx x xx The estimation of the AR arameters is done as before by taking these modified Yule-Walker equations into account (starting from the rank q + ). The estimation of the MA arameters is done first by filtering the rocess by the inverse AR filter, using the AR estimated arameters so that we come back to an MA of order q, according to the rincile in Figure 6.. Figure 6.. Princile of the estimation of MA arameters in an ARMA If the inverse AR filter is suosed erfectly known, we obtain (by using equation [6.]): = + n ( ) = n ( ) y k x k a x k n b u k n 0 q and the filtered rocess y(k) is a ure MA(q). Thus, we urely come back to a roblem of estimation of MA arameters. This model MA(q) can theoretically be modeled by an infinite order AR by writing:

11 Sectral Analysis by Stationary Time Series Modeling 6 + b z c z c z = with = c z k q n k n c = b c + δ k n 0 n We obtain a relation between the MA coefficients and the coefficients of the equivalent AR model: k q n k n c = b c + δ k δ (k) standing for the Kronecker symbol (δ (0) = and δ (k) = 0, for any k 0). In ractice, we choose to estimate an AR model of high order M such as M >> q. By using the AR arameters estimated this way, we obtain a set of equations in the form: ε k = cˆ + b cˆ [6.3] k q n k n Ideally, ε ( k ) should be null everywhere excet for k = 0 where it should be equal to. As the order of the estimated AR is not infinite, it s nothing of the sort and the estimation of the MA arameters should be done by minimizing a quadratic criterion of the form: k ε ( k ) The index k varies on a domain, which differs according to the used estimation methods. Indeed, equation [6.3] is not unlike the linear rediction error of an AR model in which the arameters would be the b n and the signal c ˆk. From this fact, the AR estimation techniques can be envisaged with, as articular cases: k = 0,..., M + q corresonding to the method of autocorrelations and k = q,..., M corresonding to the covariance method Estimation of Prony arameters The classic estimation method of the Prony model arameters is based on the recursion exression of a sum of exonentials:

12 6 Sectral Analysis which leads to: k = n n = n ( ) for [6.4] x k B z a x k k = n ( ) + n ( ) ( 0 = ) [6.5] 0 x k a x k n a u k n k N a We find the well-known equivalence between a sum of noisy exonentials and the ARMA(, ) model for which the AR and MA coefficients are identical. The oles z n are deduced from the olynomial roots: n = n = ( n) a z a z z z 0 [6.6] The classic estimation method of the AR arameters in the estimation rocedure of the Prony model arameters consists of minimizing the quadratic error: N n [6.7] e= min x k + a x k n an k= which, in a matrix form, is written as: min a n min a ( ) ( 0) x x a x + x( N ) x( N ) a x( N) Xa + x [6.8] and leads to the least-squares solution a LS : LS H H a = X X X x. [6.9] This method is sometimes called LS-Prony. When the system is solved in the total least-squares sense, we seak of TLS-Prony. We will note that the matrix X H X is the estimation of the covariance matrix, with one multilying factor ( ). This N method is thus identical to the covariance method for the estimation of the AR

13 Sectral Analysis by Stationary Time Series Modeling 63 coefficients and of the signal oles. This method can be slightly changed in the case of non-damed real sinusoids to force the estimated oles to be of unit module: min an a x* ( N + ) x* ( N) *( ) a x N x* () x* a x*0 + ( ) ( 0) x x a x ( x N ) x( N ) x( N) a [6.30] We seak then of the harmonic Prony method. Other estimation methods of the AR arameters, imlementing the correlation, can be used: LSYW and LSMYW, for examle. Once the oles are estimated ( z ˆn ) solving the Vandermonde system:, the comlex amlitudes are obtained by x( 0) zˆ zˆ zˆ B x() ˆ ˆ ˆ z z z x( ) B M M M zˆ zˆ zˆ x M VB ˆ x [6.3] Solutions in the least-squares sense and in the total least-squares sense can be envisaged. It was shown in [DUC 97] that the total least-squares solution is less efficient in terms of bias than that of the classic least-squares. On the other hand, the number of equations M of the Vandermonde system can be chosen in an otimal manner, as detailed in [DUC 95].

14 64 Sectral Analysis Order selection criteria Choosing a model system is a roblem which is as imortant as the choice of the model itself. Selecting too small an order means smoothing the obtained sectrum, while choosing too large an order introduces secondary surious eaks. There is a large number of order selection criteria which are for most cases based on the statistic roerties of the signal (maximum likelihood estimation: MLE). Others, simler and less efficient are based on the comarison of the eigenvalues of the correlation matrix to some threshold correlation matrix [KON 88]. A large number of order selection criteria use the rediction error ower decrease when the order increases. When the theoretic order is reached, this ower remains constant. However, a criterion based only on the rediction error ower shae does not make it ossible to take the estimated sectrum variance increase into account when the order is overestimated. That is why the criteria integrate these two henomena. One of the first criteria roosed by Akaike [AKA 70] was the FPE (Final Prediction Error): the estimated error corresonds to the value that minimizes: FPE k N + k N k ρ = ˆk [6.3] where: ( 0) aˆ ( l) ˆ ρ = ˆ γ + ˆ γ [6.33] k xx l xx l= k is the ower of the rediction error that decreases with k while the term N k N k increases with k (to take the estimated sectrum variance augmentation into account when k increases). The AR arameters are estimated through Yule-Walker equations with the biased estimator of the correlation. The most well known criterion roosed by Akaike is the AIC (Akaike Information Criterion) [AKA 74]: ( ˆ ρ ) AIC k = N n + k [6.34] k This criterion is more general than FPE and it can be alied by determining the order of an MA art of an ARMA model. Asymtotically ( N ) FPE and AIC are equivalent, but for a small number of samles AIC is better. It was roved that AIC is inconsistent and it tends to overestimate the order [KAS 80]. [RIS 83] roosed to modify AIC by relacing the term k by a term, which increases more +

15 Sectral Analysis by Stationary Time Series Modeling 65 raidly (deending on N) k ln (N). This criterion is named MDL (Minimum Descrition Length): ( k) = N ( ˆ ) + k ( N) MDL n ρ ln [6.35] k This criterion is consistent and gives better results than AIC [WAX 85]. It would be tedious to resent all the criteria that were develoed; for more information on this, see the following references: [BRO 85, BUR 85, FUC 88, PUK 88, YIN 87, WAX 88]. [WAX 85] exressed the AIC and MDL criteria deending on the eigenvalues of the autocorrelation matrix R ˆ y : k ˆ λt t= k+ ˆ λt AIC k = N k ln + k k k =,, [6.36] k t = k + k ˆ λt t= k+ = ( ) ln + ( ) ln ˆ λt MDL k N k k k N k t = k + [6.37] where ˆ ˆt λ ˆ t λ ˆ t + R y. It is ossible to define these criteria according to the singular values ˆt σ of the matrix Rˆ y ( M, M > ) by relacing ˆ λl by ˆ σ t in the matrices [6.36] and [6.37] [HAY 89]. λ are the ordered eigenvalues of the matrix The increase in the number of lines of the matrix R ˆ y evidently makes an imrovement of the order estimation erformances ossible. We have comared the two exressions [6.35] and [6.37] of the MDL criterion on the following examle. We consider N = 00 samles of a signal made u of two sinusoids of identical amlitudes of frequencies 0. and 0. and of one white noise. The dimension of the matrix R ˆ y is (30 x 30). The simulations were carried out for signal-to-noise ratios of 0 db and 0 db and are resented in Figures 6.3 and 6.4.

16 66 Sectral Analysis SNR = 0 db (Figure 6.3): the dimension of the subsace signal is 4 because the signal has two sinusoids. The criterion [6.37] (to curve) reaches its minimum for k = 4. However, the criterion [6.35] (bottom curve) is minimum when k = 8. These results illustrate the efficiency of the criterion defined starting from the eigenvalues of the autocorrelation matrix and the mediocre erformances of the criterion defined starting from the rediction error ower.,,,,, Calculation using the eigenvalues of Calculation using the rediction error Figure 6.3. Comarison of the MDL criteria [6.35] and [6.37] for SNR = 0 db SNR = 0 db (Figure 6.4): for a weaker SNR, the criterion [6.35] (to curve) is inefficient while the criterion [6.37] (to curve) gives a correct result. These simulations have highlighted the efficiency of the MDL criterion [6.37] and, in a more general manner, the efficiency of the criteria built starting from the eigenvalues of the autocorrelation matrix R ˆ x.

17 Sectral Analysis by Stationary Time Series Modeling 67,,,, Calculation using the eigenvalues of Calculation using the rediction error Figure 6.4. Comarison of the MDL criteria [6.35] and [6.37] for SNR = 0 db In the case of a Prony model, the model order corresonds to the number of searched exonential comonents, which evidently reduces to an order determination roblem of AR model. All the receding criteria are alicable. In the case of an ARMA model, selecting the models and q of the resective AR and MA arts is not a simle roblem. Few ublished works tackle this roblem, exceting the very simle cases. The AIC criterion is one of the most used in the form: = N ( ˆ ρ ) + ( + q) AIC,q ln q where ˆ ρ q reresents the estimated ower of the entrance noise of the ARMA model. As in the case of the AR model, the minimum of this function with two variables rovides the coule (, q) of the AR and MA orders to be taken into account Proerties of sectral estimators roduced In the case of a signal made u of a sinusoid and an additive white noise, we take interest in the AR model sectrum evolution, according to the signal to noise ratio (SNR), for a fixed AR order. Various results can be found in the literature, deending on the hyotheses taken for the modeled signal. For an examle, we can

18 68 Sectral Analysis mention [LAC 84] which rovides the equivalent bandwidth exression for a ure sinusoid with additive white noise, modeled by an AR model of order : 6 f = π ( + ) β β being the signal to noise ratio. Figure 6.5 makes it ossible to highlight that the bandwidth at -3dB of the sectrum lobe deends on the SNR: the weaker the SNR is, the more the lobe enlarges. The AR sectral resolution is affected by the noise resence. to Figure 6.5. Evolution of the AR sectrum according to the SNR in the case of a noisy sinusoid In the case of a noisy sinusoidal signal, Figure 6.6 lots the evolution of the AR sectrum according to the order of the chosen model, for a reviously fixed SNR. The larger the order is, the better the sectral resolution is (inversely related to the width of the eaks ). But if the chosen order is too big, false (or surious) eaks aear. The effect of the order choice is again illustrated on the case of a signal made u of three sinusoids in white noise. The minimal order necessary for the estimation of these three sinusoids is of 6. Of course, the resence of the additive white noise makes it necessary to choose a higher order. Figure 6.7 resents the AR sectrum evolution, while the chosen order is below 6: if we could see that like an animated cartoon, we would see the successive lines ush according to the chosen order. Below, we have reresented the eriodogram, which makes it ossible to areciate the smoothing roduced by the AR sectral analysis.

19 Sectral Analysis by Stationary Time Series Modeling 69 Figure 6.8 resents the AR sectrum evolution on the same signal made u of three sinusoids in white noise when the chosen order is suerior to 6. The three lines are better and better estimated but at the order 50, the AR modeling makes two false eaks aear, which are actually noise eaks. Figure 6.6. AR sectrum evolution according to the order of the chosen model in the case of a noisy sinusoid Figure 6.7. AR sectrum evolution (comared to the eriodogram) according to the order chosen for a signal made u of 3 sinusoids in white noise