Relation Between Spectrum Density and Wavelet Transform of Correlation Function

Home , Wavelet

Trans. of the So ciety of Instrument and Control Engineers V o l . E - 3 , No. 1 , 25 / 32 (2004) Relation between Spectrum Density and Wavelet Transform of Correlation Function

Tetsuya Tabaru ∗ and Seiichi Shin∗

This paper concerns properties of the wavelet transform of a correlation function from the viewpoint of spectrum analysis. First of all, a basic characteristic is derived as a relation between the wavelet transform and the corresponding spectrum density. The relation shows that the wavelet transform can be treated as an estimate of the spectrum density if a mother wavelet is chosen appropriately. Its bias from the true value is also obtained for this case. Moreover, a selection of the mother waveletisconsideredtoreduce the bias. This consideration provides validity of using the Gabor function as a mother wavelet for applications of the wavelet transform of the correlation function. Key Words: wavelet transform, correlation function,spectrumestimation

transform of the correlation function. In order to prove 1. Introduction the relation, we also have to investigate the case of using There are many applications of the continuous wavelet general mother wavelets. transform in various areas, for example, signal process- By the way, there is anotherproblemrelated mother ing, health monitoring, measurement and system identi- wavelets. In our preceding studies, the Gabor function fication 1), 2).Through the applications,itsusefulness is was employed as a mother wavelet to obtain the wavelet revealed widely. We focused the wavelet transform of a transform of the correlation function. Although it is em- correlation function 3) and have continued theoretical re- pirically known that use of theGabor function leads good searches and application studies, for instance, dead time results, there is no theoretical proof. If we analyze the estimation of linear systems 4), 5) and system identification relation between the wavelet transform of the correlation of a boiler plant 6). function and the spectrum density including the case of Our preceding studies showed that the wavelet trans- general mother wavelets, it should be possible to gives a form of the correlation fucntion can be treated as a kind theoretical validity of using the Gabor function. of estimate of the corresponding spectrum density if a This paper reveals the relation between the wavelet mother wavelet (analyzing wavelet) is a complex sinusoid transform of the correlation function and the correspond- (ejωpt)multiplied by a real window function, like the Ga- ing spectrum density. The derivation of the relation will bor function (also known as Complex Morlet wavelet) 3). provides the bias of the spectrum density estimated from However, there was no study about essential properties the wavelet transform of the correlation function. In addi- of the estimate, such as a bias and a variance. It is im- tion, we will prove that this bias is suppressed if a mother portant for both theory and applications to analyze these wavelet is a product of the complex sinusoid and specific properties quantitatively. window functions. That is, such a mother wavelet is ap- In addition, it is expected that there is a strong relation propriate when the continuous wavelet transform is ap- between the wavelet transform of the correlation function plied to the spectrum analysis,inparticular, if the clear and the spectrum density even when the mother wavelet relation is expected between the wavelet transform of the is not a product of the sinusoid and the window func- correlation function andthespectrum density. tion. The reason is the following: the wavelet transform This paper is organized as follows. In section 2, a ba- is very similar to the windowed Fourier transform except sic relation is derived between the wavelet transform of for frequency dependence of window functions, and the the correlation function and the spectrum density. Then, Blackman-Tukey method, which is one of the spectrum we show the condition that the wavelet transform can be estimation methods, is identical to the windowed Fourier treated as an estimate of the spectrum density and de- scribe the bias from the true value. Section 3 illustrates ∗ Graduate School of Information Science and Technology, that the bias stated above is suppressed if and only if the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656 JAPAN amotherwavelet is expressed as the complex sinusoid 26

windowed by even functions. In section 4, a numerical for the case that the wavelet transform can be regarded experiment is carried out to confirm validity of our re- as an estimate of the spectrum. sult. Section 5 concludes this paper and mentions further 2. 1 Basic Relation between Wavelet Trans- researches. form of Correlation Function and Spec- Notation: The set of real numbers will be represented trum Density by R and the set of complex numbers by C.Theaster- This subsection describes a basic analysis of a relation ∗ isk denotes complex conjugation. between the wavelet transform of a correlation function The Fourier transform of a signal x(t)isrepresented by and the corresponding spectrum density. X ω ( )anditsdefinition is the following. Let φ˜xy(a, b)denotethewavelet transform of φxy(τ), +∞ 1 −jωt which is the cross correlation function between x(t)and X(ω)= √ x(t)e dt 2π −∞ y(t). The definition of the wavelet transform directly An inner product over the space of square integrable yields functions L2(R)isdefined by φ˜xy(a, b)=ψa,b(τ),φxy(τ). (1) +∞ ∗ x, y = x (t)y(t)dt. The next theorem provides the relation between φ˜xy(a, b) −∞ and the cross spectrum density. Acrosscorrelation function between x(t) ∈ R and Theorem 1. ω0 Assume that is an arbitrary constant y(t + t ) ∈ R is denoted by φx(t),y(t+t)(τ). We employ and a cross spectrum density Φxy(ω)isn-times differen- the following definition in this paper. tiable in R.Then, T √ 1 ∗ φx(t),y(t+t)(τ)= lim x(t)y(t + t + τ)dt 2πψ (0) ω0 T →∞ T φ˜xy(a, b)= √ Φx(t),y(t+b) 2 −T a a Thus Φx(t),y(t+t)(ω), which is the Fourier transform of n−1 φx(t),y(t+t)(τ), is the cross spectrum density between x(t) + Qk + Rn, (2) and y(t + t). When t =0,theyaresimply denoted by k=1

φxy(τ)andΦxy(ω)respectively. Similarly, an auto corre- where √ +∞ x t ∈ R a ω lation function and a power spectrum density of ( ) Q (k) 0 λk ∗ ω aλ dλ. k = k Φx(t),y(t+b) a Ψ ( 0 + ) (3) are denoted by φxx(τ)andΦxx(ω)respectively. ! −∞

Amotherwavelet (or analyzing wavelet) is expressed In addition, |Rn| is bounded as follows. √ by ψ(t). In general, ψ(t)isacomplexfunction. Wavelet a |R |≤ (n) ω n max Φx(t),y(t+b)( ) bases are represented by ψa,b(t)anddeﬁned by n! ω +∞ 1 t − b n ∗ ψa,b(t)= √ ψ . × |λ ω aλ |dλ a a Ψ ( 0 + ) (4) −∞ a The parameter is the dilation parameter (or scaling pa- Proof : rameter, scale parameter) and b is the location parameter φ˜xy(a, b)=ψa,b(τ),φxy(τ) (or translation parameter, shift parameter). The wavelet transform of x(t)isdeﬁned by using these bases as fol- = ψa,0(τ − b),φx(t),y(t+b)(τ − b) lows. = ψa,0(τ),φx(t),y(t+b)(τ)

x˜(a, b)=ψa,b(t),x(t) Since inner products satisfy the following identity +∞ +∞ +∞ 1 ∗ t − b = x(t) √ ψ dt ∗ ∗ a a x (t)y(t)dt = X (ω)Y (ω)dω, −∞ −∞ −∞ In this paper, the dilation parameter a is limited to posi- we obtain tive real numbers and the domain of the shift parameter +∞ √ ∗ b is R. φ˜xy(a, b)= a Ψ (aω)Φx(t),y(t+b)(ω) dω. (5) −∞ 2. Wavelet Transform of Correlation This equation is transformed as follows by applying ω =

Function and Spectrum Density ω0/a + λ.

First, we will show a basic relation between the wavelet φ˜xy(a, b)= transform of a cross correlation function and the corre- √ +∞ ω a ∗ ω aλ 0 λ dλ Ψ ( 0 + )Φx(t),y(t+b) a + (6) sponding cross spectrum. Next, a discussion will be given −∞

The Taylor series of Φx(t),y(t+b)(ω0/a + λ)atω0/a is ﬁciently small. For Corollary 1, suppose that R in the ω ω 0 λ 0 equation (10) is so. Then Φx(t),y(t+b) a + =Φx(t),y(t+b) a √ a ω0 n−1 k √ φ˜xy(a, b) ≈ Φx(t),y(t+b) . (12) λ (k) ω0 πψ∗ a + Φ 2 (0) k! x(t),y(t+b) a k=1 This implies that the left hand of the equation can be 1 (n) ω0 regarded as an estimate of the cross spectrum density be- + Φ + ξ(λ) , (7) n x(t),y(t+b) a (1) ! tween x(t)andy(t + b). . where 0 <ξ(λ) <λfor λ>0andλ<ξ(λ) < 0forλ<0. The bias of this estimate fromthetrue spectrum density By substituting this into the equation (6), we have is +∞ √ n−1 √ ω0 ∗ φ˜xy(a, b)= aΦx(t),y(t+b) Ψ (ω0 + aλ)dλ a a √ Qk + Rn (13) −∞ 2πψ∗(0) k=1 ∞ +∞ √ 1 (k) ω0 k ∗ + a Φ λ Ψ (ω0 +aλ)dλ for the equation (2) and k! x(t),y(t+b) a k=1 −∞ √ √ ∞ a a √ +∞ a ω √ R = √ Qk (14) (n) 0 ξ λ λn ∗ ω aλ dλ. 2πψ∗(0) 2πψ∗(0) + n Φx(t),y(t+b) a + ( ) Ψ ( 0 + ) k=1 ! −∞ (8) for the equation (10). If the approximation of the equation (12) is proper, we can treat the wavelet transform of Let Rn denote the last term of this equation. Thus the the cross correlation as the estimate of the corresponding equation (2) is obtained by using spectrum density. It requires that the equation (13) or +∞ ∗ 1 ∗ ψ (0) (14) is small enough. √ Ψ (ω0 + aλ)dλ = . π a 2 −∞ In order to make (13) and (14) small, it is suﬃcient that

The upper bound of Rn can be also derived from the magnitude of Qk,deﬁned in the equation (3), is small +∞ ω for each k.Considerthetermswhichconstitute Qk.Itis (n) 0 ξ λ λn ∗ ω aλ dλ Φx(t),y(t+b) a + ( ) Ψ ( 0 + ) k ω −∞ impossible toadjust the -th derivative of Φx(t),y(t+b)( ) +∞ since it depends on the spectrum tobeestimated.In (n) ω0 n ∗ ≤ Φ + ξ(λ) |λ Ψ (ω0 +aλ)|dλ x(t),y(t+b) a contrast, the absolute value of −∞ +∞ +∞ (n) k ∗ ≤ ω |λn ∗ ω aλ |dλ. λ Ψ (ω0 + aλ)dλ (15) max Φx(t),y(t+b)( ) Ψ ( 0 + ) ω −∞ −∞ (9) can be smaller by choosing a mother wavelet ψ(t)andω0 adequately. As a result, this suppresses the magnitude of

Qk.Furtherdiscussions about these choices will be shown When Φxy(ω)isinfinitely differentiable, the following in the next section. result is obtained similarly. As stated above, we can make the bias smaller for an Corollary 1. Assume that ω0 is an arbitrary constant. unknown object by reducing the magnitude of the equa- If Φxy(ω)isinfinitely differentiable, √ |Qk| ∗ tion (15). However, to suppress ,itisalsonecessary 2πψ (0) ω0 φ˜xy(a, b)= √ Φx(t),y(t+b) + R, (10) that the magnitude of a a (k) ω0 where Φx(t),y(t+b) a (16) ∞ is not too large. This brings another problem. The mag- R = Qk b k=1 nitude is dependent on the location parameter .Itcan jωb ∞ √ +∞ be seen from Φx(t),y(t+b)(ω)=e Φxy(ω). The deriva- a (k) ω0 k ∗ = Φ λ Ψ (ω0 +aλ)dλ ω k! x(t),y(t+b) a tive and highorderderivatives of Φx(t),y(t+b)( )maybe k=1 −∞ very large depending on the value of b.Hencewemustbe (11) aware thatnotall b allow use of the approximation (12). The range of b whichsuppress the magnitude of (16) de- 2. 2 Spectrum Density Estimation by Wavelet pends on the object to be estimated. Therefore we don’t Transform of Correlation Function and Its Bias (1) This result matches the one of the article 3), though the difference of the Fourier transform√ definition causes the Suppose that Qk and Rn in the equation (2) are suf- difference of the constant multiplier 2π. 28

discuss it anymore and leave it for further studies. For the where w(t) ∈ R is an even function. Let us call ωp the analyses in the rest of the paper, the location parameter center frequency of the mother wavelet. b is assumed to be suchthatthe equation (16) is not too If a function w(t)isarealand even function,itsFourier large. transform W (ω)isalsorealand even. From the equation (18), the Fourier transform of ψ(t)satisﬁes 3. Analysis for Windowed Complex Si- nusoid Wavelets Ψ(ω)=W (ω − ωp). (19)

Therefore Ψ(ω)issymmetrical about ω = ωp and ωp is This section will focus mother wavelets which can be ex- the center frequency of the mother wavelet. Also note pressed as a product of a even function w(t)andacomplex that Ψ(ω)isrealfunction. sinusoid, i.e. The Gabor function 3), 8) (Complex Morlet wavelet) is ψ t w t ejωpt. ( )= ( ) (17) one of the typical this type mother wavelets. If the Ham- We shall call these wavelets the windowed complex sinu- ming or Hanning window, commonly used in the spec- 9) soid wavelets in this paper. The main purpose of this trum analysis ,arechosen as w(t), the mother wavelet section is to advance the analysis stated in the previous also becomes this type wavelet. 2 section for the wavelets. When w(t)=e−t ,thewavelet It is desirable to choose a positive real function as Ψ(ω) is known as the Gabor wavelet (Complex Morlet wavelet). when the wavelet transform ofacorrelation function is It is employed often in wavelet analysis applications and treated as an estimate of a spectrum density. This can its usefulness is confirmed experimentally. be understood from the equation (5), which shows that Section 2. 2 illustrated that the wavelet transform of a φ˜xy(a, b)isaweightedmovingaverage of the spectrum ∗ cross correlation function can be considered as an estimate density Φx(t),y(t+b)(ω)andΨ (ω)isitsweighting function. of the corresponding cross spectrum density (Eq. (12)) if Thus negative Ψ(ω)maycausetroubles, for example, an the bias of the equation (13) or (14) is small. To do this, a estimated power spectrum becomes negative and an esti- mother wavelet ψ(t)mustbeselected appropriately. We mated cross spectrum has strange phase values. For the will show that the windowed complex sinusoid wavelets above reason, it is better to use positive real Ψ(ω). are proper for this purpose. That is to say, if the wavelets 3. 2 Validity of Definition of Center Frequency are employed, the wavelet transform of the cross correla- It is possible to justify the choice of the center frequency tion is equivalent to the estimate of the cross spectrum. in the section 3. 1 from another viewpoint, that is, the In addition, its inverse is also true. If a mother wavelet choice also suppresses the bias in the section 2. 2. Con- is chosen such that |Qk| is small in a sense and the bias sider the integration +∞ is suppressed, it becomes a windowed complex sinusoid k ∗ λ Ψ (ω0 +aλ)dλ (20) wavelet. This implies that it is desirable to select the −∞ windowed complex sinusoid as the mother wavelet if the included in the equation (3). It can be proven that for connection is emphasized between the wavelet transform each k, ω0 which minimizes the magnitude of the integra- of the correlation function and the spectrum density. tion is coincident with the center frequency ωp defined in First, we explainthewindowed complex sinusoid the section 3. 1. Suppressing the integral (20) makes the wavelet and define its center frequency. Next, validity considered bias small since the sum of Qk are proportional of the definition is shown from the viewpoint that it sup- to the equation (13) and (14), which are the bias when press the bias considered in the section 2. 2. The proof the wavelet transform of a correlation function is treated of the validity will also reveal that the wavelets reduce as an estimate of a spectrum density. We will show the the bias. Next, we will show that if a mother wavelet is result as the following theorem. chosen such that the bias is suppressed, then it becomes Theorem 2. For the mother wavelet of the equation awindowed complex sinusoid wavelet. Finally, a relation (17), ω0 such that minimize with the Blackman-Tukey method will be discussed. +∞ k ∗ 3. 1 Windowed Complex Sinusoid Wavelet and Ik(ω0)= λ |Ψ (ω0 +aλ)|dλ (21) −∞ Center Frequency is the same as the center frequency ωp in the section 3. 1. As stated before, the windowed complex sinusoid Proof:Fromtheequation (19), mother wavelets are represented by +∞ k ∗ jω t Ik(ω0)= λ |W (ω0 − ωp + aλ)|dλ . (22) ψ t w t e p , ( )= ( ) (18) −∞ 29

Now let ω = ω0 − ωp and define Ik(ω)=Ik(ω0 − ωp). The equality holds if ω =0,i.e. ω0 = ωp.Henceω0 = ωp I ω Then we obtain minimizes k( 0). +∞ k ∗ We chose ωp in the equation (17) for the center fre- Ik(ω)= λ |W (ω + aλ)|dλ . (23) −∞ quency of the mother wavelets. As seen above, this re- Apply the transformation λ = ω + aλ to this equation duces Qk and Rn of the equation (2) and R in the equation and let us reuse λ to represent λ after the translation. (10) and, in consequence, the bias is suppressed when the Since a>0isassumed, thus wavelet transform of the correlation function is treated as +∞ λ − ω k dλ an estimate of the spectrum density. Also in this sense, I ω |W ∗ λ | k( )= a ( ) a −∞ the choice has validity. +∞ The integration (20) coincides with (21) if Ψ(ω)ispos- 1 k ∗ = (λ − ω) |W (λ)|dλ . (24) ak+1 itive. Hence, ω0 such that minimize −∞ +∞ i) k is odd: It is trivial that Ik(ω)iszeroandminimum k ∗ λ Ψ (ω0 +aλ)dλ (32) at ω =0,thatis,whenω0 = ωp.Thisiseasily derived by −∞ the fact that the integrand is odd function. is the center frequency ωp defined in the section 3. 1. k+1 ii) k is even: Let us calculate a Ik(ω)first.Whenk 3. 3 Windowed Complex Sinusoid Wavelet and is even, the integrand is always positive. It follows that Bias 0 As we saw in the proof of Theorem 2, the equation (20) ak+1I ω λ − ω k|W ∗ λ |dλ k( )= ( ) ( ) k Q −∞ becomes zero if is odd. This implies that k of the +∞ equation (2) disappear for odd k.Thewindowed complex k ∗ + (λ − ω) |W (λ)|dλ . (25) 0 sinusoid wavelets suppress the bias in this sense, too. Apply the transformation λ =−λ to the first integration Moreover, its inverse also holds. Suppose that a mother Q k i.e. and recall that k is even and W (ω)isevenfunction. Then wavelet is chosen so that k is zero for odd , +∞ 0 2l+1 ∗ k+1 k ∗ λ Ψ (ωp + aλ)dλ =0,l=0, 1, ···. (33) a Ik(ω)= (−λ − ω) |W (−λ )|d(−λ ) −∞ +∞ +∞ Then, the wavelet becomes a windowed complex sinusoid. λ − ω k|W ∗ λ |dλ + ( ) ( ) (26) Theorem 3. The equation (33) is satisfied and Ψ(ω) 0 +∞ is real if and only if ψ(t)isthewindowed complex sinusoid λ ω k|W ∗ λ |dλ = ( + ) ( ) mother wavelet. 0 +∞ Proof: k ∗ + (λ − ω) |W (λ)|dλ (27) (Sufficiency) It is already shown in the section 3. 1 that 0 ω ψ t +∞ Ψ( )isrealif ( )isthewindowed complex sinusoid k k ∗ = ((λ + ω) +(λ − ω) )|W (λ)|dλ . mother wavelet. The proof of Theorem 2 also shows that 0 the equation (33) holds under the same condition. (28) (Necessity) Let us define Ψe(ω)andΨo(ω)as ak+1I We can calculate k(0) similarly. 1 Ψe(ω)= (Ψ(ω)+Ψ(−ω +2ωp)) (34) +∞ 2 ak+1I λk|W ∗ λ |dλ k(0) = 2 ( ) (29) 1 0 Ψo(ω)= (Ψ(ω) − Ψ(−ω +2ωp)) . (35) 2 Since k is even, These are defined to satisfy the following equations. k/2 k k k 2l k−2l (λ + ω) +(λ − ω) − 2λ =2 kC2lλ ω Ψ(ω)=Ψe(ω)+Ψo(ω)(36) l=1 Ψe(ωp + ω)=Ψe(ωp − ω)(37) ≥ 0. (30) Ψo(ωp + ω)=−Ψo(ωp − ω)(38) Integrands in both the equations (28) and (29) are always ω positive. Therefore It means that Ψ( )isdecomposedintotwocomponents ω ω Ψe( )andΨo( ), and the formar is symmetric about Ik(ω) − Ik(0) ω ω = p and the latter is anti-symmetric. +∞ k/2 1 C λ2lωk−2l|W ∗ λ |dλ Since the above decomposition gives = k+1 2 k 2l ( ) a +∞ 0 l=1 λ2l+1 ∗ ω aλ dλ ≥ . Ψe ( p + ) =0 0 (31) −∞ 30

+∞ 2l ∗ (the equation (40)). This fact yields the next corol- λ Ψo(ωp +aλ)dλ =0, −∞ lary. the following equations are satisfied. Corollary 2. Assume that l is natural number and +∞ 2l+1 ∗ We(ω) =0.Foragiven window function w(t)such λ Ψ (ωp +aλ)dλ −∞ that its Fourier transform isn’t even function, there ex- +∞ w t 2l+1 ∗ ists another window function ( )suchthat the same = λ Ψo(ωp +aλ)dλ (39) −∞ Q2l, but Q2l+1 =0,anditsFouriertransform satisfies +∞ W ω We ω ln ∗ ( )= ( ). λ Ψ (ωp +aλ)dλ −∞ The corollary states that we can select a mother wavelet +∞ and the relatedwindow function such that Qk =0for λ2l ∗ ω aλ dλ = Ψe ( p + ) (40) odd k,without any other effects from the viewpoint of −∞ the bias. In other words, letting Qk =0foroddk doesn’t The equations implythatthe equation (33) holds if Ψo(ω) alter the value of Qk for even k. is set to zero. Thus, itissufficienttoletΨ(ω)=Ψe(ω) and choose ψ(t)satisfying Consequently, the windowed complex sinusoid is a desirable mother wavelet to relate the wavelet transform of the Ψ(ωp + ω)=Ψ(ωp − ω). (41) cross correlation function with the cross spectrum density. Next, consider ψ(t)whichsatisfies this equation. From 3. 4 Relation with Blackman-Tukey Method the inverse Fourier transform, This subsection will treat the relation between two esti- +∞ 1 jωt ψ(t)= √ Ψ(ω)e dω mates of a cross spectrum density: one is calculated from π 2 −∞ awavelet transform of a cross correlation function and an- +∞ 1 j(ω+ωp)t other is by the Blackman-Tukey method. In particular, = √ Ψ(ω + ωp)e dω 2π −∞ we will discuss their biases mainly. +∞ w t 1 jωpt jωt The window function treated in this section, ( ), is = √ e Ψ(ω + ωp)e dω. (42) 2π −∞ quite similar to the one in the Blackman-Tukey method Now, let us suppose that if the condition w(0) = 1 is added. The only difference is +∞ that w(t)usedintheBlackman-Tukey method must sat- w t √1 ω ω ejωtdω. ( )= Ψ( + p) (43) M> 2π −∞ isfy one more condition: there exists a 0suchthat jω t w τ τ |τ| >M w Then, ψ(t)=w(t)e p .Wewill show that w(t)becomes ( )=0for satisfying .Since (0) = 1 means ∗ w −t w t ψ (0) = 1, the equation(12) becomes simpler as follows. even function, that is, ( )coincideswith ( ). From √ a ω the previous equation, ˜ p √ φxy(a, b) ≈ Φx(t),y(t+b) (44) +∞ 2π a w −t √1 ω ω ejω(−t)dω. ( )= Ψ( + p) Suppose that a mother wavelet is the windowed com- 2π −∞ w ω −ω plex sinusoid wavelet and it also satisfies (0) = 1. Then, By using the transformation = and the equation √ π w ω (41) , we have Q − √2 (0) p . 2 = a2 Φx(t),y(t+b) a (45) −∞ 2 a 1 jωt w(−t)= √ Ψ(−ω + ωp)e dω π From this and the equation (2), we obtain 2 +∞ √ a ω +∞ ˜ p 1 jωt √ φxy(a, b)=Φx(t),y(t+b) = √ Ψ(−ω + ωp)e dω 2π a 2π −∞ 1 w (0) ωp +∞ − Φx(t),y(t+b) + ···. 1 jωt 2 a2 a = √ Ψ(ω + ωp)e dω 2π −∞ (46) w t . = ( ) Here, let us consider the spectrum density estimate cal- Hence w(t)isevenfunction. culated from the wavelet transform of the corresponding On the other hand, if Ψ(ω)isarealfunction, that cross correlation function. The equation implies that the is when W (ω)isreal(recallW (ω)=Ψ(ω + ωp)), then bias of the estimate is inversely proportional to the square w(t)=w∗(−t). From this and the fact that w(t)iseven, of the width of the wavelet basis since the width is pro- w(t)isarealfunction. portional toadilationparameter a in time domain.

As seen in this proof, Qk depends on only Wo(ω)for This result is almost the same as the one of the 9) odd k (the equation (39)), and only We(ω)foreven k Blackman-Tukey method , i.e. the bias isproportional 31 /M 2 M to 1 ,where is the width of the window function. 10 The only diﬀerence is that the window width M is constant for the Blackman-Tukey method while the width is 5 dependent on frequency for the wavelet case.

4. Numerical Experiment 0

This section illustrates the above results by a numerical −5 experiment. We estimate a powerspectrumdensityfrom the wavelet transform of a auto correlation function and −10 consider the experiment result. To generate a test signal, a white noise of 0 (dB) was

Magnitude (dB) −15 passed through the filter whose transfer function was 1 G(s)= . (47) s2 +0.7s +1 −20 The sampling rate was set to 0.5 second and the number of samples was set to 4096. We calculated the wavelet trans- −25 form of the auto-correlation function of the test signal and estimated its power spectrum density from the equation −30 (12). The location parameter b is fixed to 0 toobtainthe 0.1 1.0 Frequency (rad/sec) power spectrum density. We selected the Gabor function for a mother wavelet. It is expressed as follows. Fig. 1 Power spectrum estimation using the wavelet trans- 2 form of a correlation function. The thick line is the ωp 2 ψ(t)=exp − t exp(jωpt)(48) 2γ2 estimated power spectrum by the wavelet transform of the auto-correlation function of a signal. It is ob- In this experiment, ωp and γ were set to 1(rad/sec) and tained by the equation (12). The thin line is its true 2π respectively. To simplify the calculation, the mother value. The difference between them is small. This result shows that it is possible to estimate the spectrum ψ∗ wavelet was multiplied by a constant so that (0) = 1. density by the wavelet transform of the correlation This function is obtained by using the Gaussian window function. The dashed line is the plot of (50), which is the true value plus the first term of the estimate’s as follows. 2 bias. The difference becomes smaller. ωp 2 w(t)=exp − t (49) 2γ2 Figure 1 shows the caluculated result. The thick line spectrum density. It is desirable to use the windowed com- jω t is the power spectrum estimated by the above way. The plex sinusoid wavelet, which is a complex sinusoid e p thin line is G(jω) 2,thatis,thetruevalueofthespec- multiplied by a real even windowfunction,ifweexpect trum. You can see that both lines are very similar each to relate the wavelet transform of the correlation function other though there is a small bias between them. As this with the spectrum density. The Gabor function is one of example, a spectrum density can be estimated from the the commonly used wavelets of them. As mentioned in wavelet transform of a correlation function if an appro- the introduction, we employed the wavelet transform of priate mother wavelet are chosen. the correlation function fordeadtimemeasurementand Moreover, we calculated the true value of system identification. The result theoretically justifies use ωp 1 w (0) ωp of the Gabor function for these studies sincetheyhavea Φxy − Φxy , (50) a 2 a2 a close relation with the spectrum density. It has great sig- which is the first two terms of the right hand of the equa- nificance for the related application researches. Also for tion (46). The value is plotted by the dashed line in general applications using the continuous wavelet trans- Fig. 1. It can be seen that the difference becomes smaller form of the correlation function, we recommend to start by adding the second order term. their developments with such mother wavelets. They en- 5. Conclusion able us to link the wavelet transform with the notion of frequency and spectrum, and it will be help to understand This paper considers the relation between the wavelet the connection with conventional methods. We consider transform of a correlation function and the corresponding that it is desirable and effective at the beginning of the 32

development since many properties of the object are unknown initially. Tetsuya T ABARU (Member) There are some further researches. We have to consider He received the B.E. and M.E. Degrees re- the center frequency of wavelets quantitatively for general spectively in 1992 and 1994. He also received Doctor of Engineering from the University of mother wavelets. It is necessary to investigate a range of Tokyo in 2003. Since 1995, he has been a Re- the locationparameter such that the discussion of this pa- search Associate at the University of Tokyo. per is still valid. The evaluation of the variance is left in His current research interests include continuous wavelet analysis and its application, system spite of its importance. Another method was proposed to identification, signal processing. estimate a spectrum density based on the wavelet trans- 10), 11) form of a correlation fuction .Themethod calcu- Shin SEIICHI (Member) lates a periodogram directly from the wavelet transform He received B.E. and M.E. Degrees respec- of a signal. It may be necessary to study the properties tively in 1978 and 1980. He also received Doc- of the estimates by such a method. tor of Engineering from University of Tokyo in 1987. He was awarded Paper Award from the References Society of Instrumentation and Control Engi- neers in 1991, 1992, 1993 and 1998 including 1) R. Ashino and S. Yamamoto: Wavelet Analysis, Kyoritsu the Takeda Prize. He is a Fellow of SICE. Shuppan (1997) (in Japanese) 2) Mini Special Issue — Practical Applications of Wavelet Transform, Journal of SICE, 39-11, 679/722 (2000) (in Japanese) 3) T. Tabaru: Wavelet Analysis and Correlation Analysis, Journal of SICE, 39-11, 683/688 (2000) (in Japanese) 4) T. Tabaru and S. Shin: Deatd Time Measurement Based on Wavelet Analysis of Correlation Data, Trans. of SICE, 35-3, 357/362 (1999) (in Japanese) 5) T. Tabaru and S. Shin: Reconsideration of Dead Time Measurement by Wavelet, Proceedings ot the 38th SICE Annual Conference, 695/696 (1999) (in Japanese) 6) K. Nakano and Y. Toyoda: Wavelet-Based Identification of Time-Delay Systems and Its Application to Drum-Boiler Systems, Journal of the Society of Instrument and Control Engineers, 39-11, 701/705 (2000) (in Japanese) 7) K. Nakano, T. Tabaru, S. Shin, Y. Toyoda, T. Tsujino and T. Sanematsu: Wavelet-Based Identification for Control of Watertube Drum Boilers, Transaction of IEE Japan, 122- D-2, 111/119 (2002) (in Japanese) 8) M. Sato: Mathematical foundation of wavelets, The Jour- nal of the Acousticial Society of Japan, 47-6, 405/421 (1991) (in Japanese) 9) S. Sagara, K. Akizuki, T. Nakamizo, T. Katayama: System Identification, SICE (1981) (in Japanese) 10) K. Takada and K. Nakano: On Estimation of Transfer Functioon Models via Wavelet-Spectrum, Proceedings of SICE 1st Annual Conference on Control Systems, 365/370 (2001) (in Japanese) 11) K. Takada and K. Nakano: On Estimation of Transfer Function Models via Wavelet-Spectrum, The Papers of Technical Meeting on Industrial Instrumentation and Con- trol (IIC-02-47), 13/18 (2002) (in Japanese) 12) T. Tabaru and S. Shin: Relation between Wavelet Trans- form of Correlation Function and Spectrum Density, The Papers of Technical Meeting on Industrial Instrumentation and Control (IIC-02-48), 19/24 (2002) (in Japanese)