Estimation and detection of a periodic signal

Transcription

1 Estimation and detection o a periodic signal Daniel Aronsson, Erik Björnemo, Mathias Johansson Signals and Systems Group, Uppsala University, Sweden, Daniel.Aronsson,Erik.Bjornemo,Mathias.Johansson}@Angstrom.uu.se Abstract. We consider detection and estimation o a periodic signal with an additive disturbance. We study estimation o both the requency and the shape o the waveorm and develop a method based on Fourier series modelling. The method has an advantage over time domain methods such as epoch olding, in that the hypothesis space becomes continuous. Using uninormative priors, the noise variance and the signal shape can be marginalised analytically, and we show that the resulting expression can be evaluated in real time when the data is evenly sampled and does not contain any low requencies. We compare our method with other requency domain methods. Although derived in various dierent ways, most o these, including our method, have in common that the so called harmogram plays a central role in the estimation. But there are important dierences. Most notable are the dierent penalty terms on the number o harmonic requencies. In our case, these enter the equations automatically through the use o probability theory, while in previous methods they need to be introduced in an ad hoc manner. The Bayesian approach in combination with the chosen model structure also allow us to build in prior inormation about the waveorm shape, improving the accuracy o the estimate when such knowledge is available. Keywords: Periodic signal, Bayesian inerence. INTRODUCTION The problem o detecting and estimating a periodic signal oten arises in many scientiic ields. In this paper we consider a general model or a periodic signal, (t) = c + A h cos(hωt) + which is measured in additive white noise at discrete time instants: B h sin(hωt), () d n = (t n ) + e n, n =...N () is the number o harmonics that we wish to include in our model. For uniorm sampling (t n = nt s or some sampling interval T s ) the Nyquist theorem applies so that ω/π < /T s or aliasing not to occur. In the extreme case, when there is no common divisor between the sampling times t n, the aliasing phenomenon vanishes and there is no limit or how large can be. In practice however, sampling times can only be determined to some inite precision. One may then say that there exists a common An alternative model, motivated by uncertain sampling times, in which data are grouped in time bins has been studied by Gregory and Loredo [].

2 time unit T s so that t n = k n T s or dierent integers k n, which ultimately imposes an upper limit on []. In this paper we will regard as a design parameter. When the data is uniormly sampled, an alternative approach is to vary with ω so that it always takes the maximum value that conorms to the Nyquist theorem.. OPTIMAL FREQUENCY ESTIMATION The objective here is to produce the posterior or the undamental requency ω, p(ω D,I) (More parameters may be added to the right o the conditioning bar depending on what inormation is available). The problem is usually reormulated to a search or the sampling distribution p(d ω,i) where D = d n }. The posterior or ω is then given, up to a normalisation constant, by multiplying p(d ω, I) with whatever prior or ω one preers. Usually, Jerey s prior is used. This is motivated by noting that we might iner the period time instead o the requency; we then wish to make our inerence invariant with respect to this arbitrary choice. We start by assuming a known noise power and obtain the sampling distribution by marginalising over the unknown amplitudes. By the maximum entropy principle we model the noise as Gaussian white noise with expectation zero and variance σ. This is motivated i we know the noise to have zero mean and ixed power. Denoting the amplitudes by a = [c A B A...] T and assigning Gaussian priors to them, we have p(d ω,σ,i) = p(da ω,σ,i)da, (3) where the integrand can be expressed as p(d a,ω,σ,i)p(a R a,i) exp σ N n= (d n (t n )) }exp at R a a }. (4) R a is the covariance matrix or the amplitudes. It is considered to be known throughout this paper. By varying its elements, one may incorporate dierent prior knowledge into the model. For example, i it is known that the signal has a dominant undamental then the elements corresponding to A and B should be assigned higher values than the rest. Writing also the irst exponent on vector orm, we have p(d a,ω,σ,i)p(a R a,i) exp ( σ N where d = N n= d n/n and d d T a + a T R ) a } at R a a, (5) The normalisation constant is (πσ ) N/ (π) (+)/ R a

3 d T = [d n Nd d n cosωt n R(ω) d n sinωt n I(ω) d n cosωt n R(ω)...] (6) and R = N cosωt n sinωt n cosωt n cosωt n cosωt n cosωt n cosωt n sinωt n cosωt n cosωt n sinωt n sinωt n cosωt n sinωt n sinωt n sinωt n cosωt n cosωt n cosωt n cosωt n cosωt n sinωt n cosωt n cosωt n (7) The sums are taken over n =,...,N. In order to marginalise over the a we diagonalise the covariance matrix: σ R + R a = VDV = VDV T (8) D is diagonal and consists o the eigenvalues λ} o the covariance matrix. V are the orthonormal eigenvectors. By choosing a new set o variables b = V T a, (5) transorms into p(d a,ω,σ,i)p(a R a,i) exp N d } σ + σ dt Vb bt Db ( = exp N d ) σ + b (d T V) σ b λ + b (d T V) σ b λ (9) (x) n indicates the n:th elements o the vector x. (9) obviously actors in the elements o b which allows us to carry out the marginalisation by using the identity exp( αx + βx)dx = π/α exp(β /4α). For numerical reasons, it is usually a good idea to calculate the logarithm o p(d ω,σ,r a,i). We now have log p(d ω,σ,r a,i) = log p(d a,ω,σ,i)p(a R a,i)da = N logπσ + logπ log R a + log π N d h= λ h σ + h= (d T V) h σ 4 λ h () This gives us a general procedure or ridding the nuisance parameters a when the noise variance σ and the amplitude covariance R a are known. It is common to subtract the mean value rom the original data series and assume that c = (hence it is implicitly assumed that c = d). It will not suice to merely set the irst element in R a to zero, because the matrix will then become singular. Instead

4 we remove the irst row and column rom R a and R and correspondingly the irst element rom d and a. In the ollowing exposition we will assume that these operations have been perormed. One may consider a number o assumptions:. Assume uniormly sampled data, that is t n = nt s.. Neglect low requencies (ωt s >> ). Then, in combination with assumption, N/ I. This is usually called the narrowband approximation. R 3. Assign independent priors to the amplitudes. R a = / diag[δ, δ, δ,...] 4. Assign uniorm priors to the amplitudes, that is let δ h. In order or () to simpliy to an analytic expression in ω, σ and R a, we require R /σ + R a to be diagonal. This requires us to make assumptions 3 above. We then have σ R + R a and V = I [ N = diag σ + δ, λ N σ + δ, λ N σ + ] δ,... = D λ 3 () This yields the likelihood given assumptions 3: p(d ω,σ,δ},i) = (πσ ) N/ (π) R a π N/σ + /δ h } exp N d σ + NC(hω) σ 4 (N/σ + /δh ). () ere we have used the power spectrum C(ω) (R (ω) + I (ω))/n, where R(ω) and I(ω) are deined in (6). The power spectrum is eiciently calculated or uniormly sampled data by using the ast ourier transorm. The requency estimation may thereore be evaluated in real time when assumptions 3 are used. Unortunately, σ cannot be removed analytically rom the above expression. To allow or analytic marginalisation we must also make assumption number 4. Letting the δ h } go to ininity and marginalising over the amplitudes we get p(d ω,σ,i) = [ (/N) (πσ ) N/ R a σ exp σ N d C(hω) ]} (3)

5 We may now apply Jereys prior to σ and use σ K exp( α/σ )dσ = α K/ Γ(K/)/, which demands α > and K N +. ere, K = N : p(d ω,i) (/N) (π) N/ R a Γ This is the likelihood given assumptions 4. ( N ) [ N d C(hω) ] N (4) 3. TE SIGNAL SAPE We estimate the shape o the periodic signal by inding the expectation o the amplitudes a. We begin by considering the case where all variances and also the requency ω are known, that is we look or E(a D,ω,σ,R a,i) = ap(a D,ω,σ,R a,i)da (5) Since the posterior or a is Gaussian we may just as well look or its peak which is the same as searching or the peak o (9). It does not matter whether we consider a or b since they have Gaussian distributions and are linearly related. Dierentiating with respect to b n and equating with zero we get E(b n D,ω,σ,R a,i) = (dt V) n λ n σ. (6) By writing this on vector orm and making a linear transormation we get the expected values or a: E(b D,ω,σ,R a,i) = σ D V T d E(a D,ω,σ,R a,i) = σ VD V T d (7) The signal may now easily be reconstructed: est( (t)) = σ [cosωt sinωt cosωt...]vd V T d. (8) In many situations it is reasonable to estimate the waveorm shape only ater one has decided on a value or ω. This motivates regarding ω as known in the above expressions. owever, one may want to remove the noise variance σ. When assumption 4 is made, the marginalisation using Jerey s prior may be carried out analytically (except or the eigenvalue decomposition), but this is not o much use since marginalisation over the uncertainty does not change the expected value. ence, also ater removing σ the expectations (7) will remain the same when uniorm priors on the amplitudes are used. The marginalisation over σ needs to be perormed numerically i R a. The corresponding integral is easily approximated by a sum, although the summation process may be time consuming since an eigenvalue decomposition needs to be carried out or each value o σ.

6 It may sometimes be motivated also to marginalise over ω. This is necessarily done numerically or by using certain approximations. See the next section or comments on this. 4. SIGNAL DETECTION The question o whether a periodic signal is present or not is answered by calculating the ratio p( periodic model D,I) p(d periodic model,i) p( periodic model I) = p( alternative model D,I) p(d periodic model,i) p( alternative model I). (9) The prior probabilities or the respective models are usually taken to be the same, reducing the detection problem to just comparing the probability or data or the dierent models. The alternative model that we will consider here is one where data are taken to be just white Gaussian noise. Using Jerey s prior on the noise variance, the probability or data is then easily ound to be p(d white noise, I) = Γ(N/)( d (πn) N/ ) N/. () For our periodic model, we have previously seen that the amplitudes a may be removed in the general case i their prior covariance matrix is given: π p(d ω,σ,r a,i) = (πσ ) N/ (π) exp N } d R a h= λ h σ + (d T V) h h= σ 4. λ h () We then apply Jerey s prior to ω and σ and marginalise to ind the probability o D given our periodic model: p(d periodic model, I) = p(d ω,σ,r a,i)dωdσ () The marginalisation needs to be evaluated numerically. This can be done by approximating the marginalisation with a double sum. An alternative approach suggested by Bretthorst [3] is to given σ it a Gaussian unction in ω to the sampling distribution () over which marginalisation then can be perormed analytically. owever, caution needs to be taken in the present context, since the distribution or ω usually has several sharp peaks. It may thereore be a better idea to pick out the highest peaks o p(d ω,σ,r a,i) (given a certain σ) and approximate it with a sum o Gaussians. Note that this should be done by itting parabolas to the logarithm (Eq. ) or numerical reasons. The same approximation should be applied also i the special cases () or (4) are used.

7 5. COMPARISONS WIT OTER ESTIMATORS Many solutions to the problem o the detection o a periodic signal have been proposed over the years, but relatively ew o those include the estimation o the undamental requency. The common ad hoc solution is to simply take the maximum o the power spectrum C(ω) (given uniormly sampled data). inich [4] deined the harmogram and used its maximum value to ind the most probable requency. With our notation this conorms to est(ω) = max ω C(hω). (3) It is thereore interesting to see that the harmogram appears in the special cases () and (4). owever, probability theory automatically introduces penalty terms in our expression which avour simple models (low ) beore complicated ones (high ). 6. RESULTS AND COMMENTS In this paper we derive a scheme or detecting a periodic signal in noisy measurements and or estimating its shape and undamental requency. Certain simpliications yield analytic expressions or these estimates. The algoritm can resolve requencies ar beyond the capability o the power spectrum, which is the prevalent ad hoc method or estimating periodic components. Figure shows a short data sequence containing about two periods o a voice recording. As is clear rom the igure, the present method resolves the correct requency while the power spectrum does not. Our method also handles non-uniormly sampled data in noise o unknown power. Moreover, it is possible to incorporate prior knowledge about the waveorm shape when such inormation exists. The number o harmonics to use in the model or the periodic component is a design parameter, but the algorithm automatically punishes unnecessarily high values on. REFERENCES. P. C. Gregory, and T. J. Loredo, The Astrophysical Journal (99).. G. L. Bretthorst, Maximum Entropy and Bayesian Methods in Science and Engineering (3). 3. G. L. Bretthorst, Bayesian Spectrum Analysis and Parameter Estimation, Springer-Verlag, M. J. inich, IEEE Transactions on Acoustics, Speech, and Signal Processing (98).

8 . Signal amplitude power t[s] Power spectrum log probability [z] log likelihood or ω FIGURE. A short data sequence containing about two periods o a voice recording is displayed. As is clear rom the igure, the method presented in this paper resolves the correct requency while the power spectrum does not. The log likelihood or ω in the bottom igure is unnormalised. [z]