Single-tone Frequency Tracking Using a Particle Filter ...

Single-tone Frequency Tracking Using a Particle Filter with Improvement Strategies Bin Liu1,3, Chunlin Ji2, Xiaochuan Ma1, and Chaohuan Hou1, Fellow, IEEE Institute of Acoustics, Chinese Academy of Sciences, Beijing, 100190, China 2 Department of Statistical Science, Duke University, Durham, NC 27708-0251, USA 3 Graduate University, Chinese Academy of Sciences, Beijing, 100190, China E-mail: [email protected], [email protected], {maxc,hch}@mail.ioa.ac.cn 1

filter (EKF) frequency tracker has been devised [6]. EKF usually leads to non-stable even divergent estimate in cases where nonlinearity or non-Gaussian exists. However, nonlinear models are involved in the frequency tracking problem. So the Sequential Monte Carlo methods, also known as the particle filters, will be good candidates for solving this problem. In [7], a specific unscented particle filter (UPF) has been applied in tracking of time-frequency components. In this paper, we focus on the problem of frequency tracking for a non-stationary single-tone sinusoidal signal in Gaussian noisy environment. A nearly constant frequency (NCF) model, which is adapted from the target tracking discipline, is presented for describing the evolution of the time varying frequency. A specific particle filter embedded with some improvement strategies, i.e., a Gaussian Kernel based regularization and a Metropolis-Hastings based MCMC move step, is utilized for this problem. This paper is organized as follows. In Section 2, we present the NCF model. In Section 3, we propose the Monte Carlo sampling algorithm designed for this problem. In Section 4, we present some simulation results before concluding the paper in Section 5.

Abstract This paper investigates a robust method for extracting frequency online from a noisy sinusoidal signal. A nearly constant frequency (NCF) model, which is adapted from the target tracking discipline, is presented to describe the evolution of the time varying frequency. A particular particle filtering algorithm, called bootstrap filter, is improved with a Gaussian kernel based regularization and a Metropolis-Hastings based Markov Chain Monte Carlo (MCMC) technique, for solving this problem. Some representative scenarios are designed for tests. The results of the simulation using synthetic data show the proposed method’s efficiency and superiority to some existing methods.

1. Introduction In audio signal processing, often we need to track the evolution of frequency components with respect to time so as to follow the pitch of a piece of music. The problem of frequency estimation and tracking also has numerous applications in other science and engineering problems [1] . For stationary deterministic constant sinusoidal parameters, numerous frequency estimators, which include periodogram [2], maximum likelihood [2], nonlinear least squares, notch filtering and subspacebased techniques, etc., have been proposed. Some of theses batch-mode estimators have been modified to the online mode, such as the least mean squares [3] as well as recursive least squares [4] realizations of several linear prediction schemes and the adaptive notch filter [5], for non-stationary frequency tracking. Apart from them, a state-space based extended Kalman

2. NCF model We consider the problem of tracking the timevarying frequency of a real sinusoid in additive zeromean Gaussian noise. To begin with, we define a continuous-time signal model as follow: yt = st + nt = α t cos(θ t ) + nt (1) where t is a continuous time variable, st is the true signal with amplitude α t and angle θ t , and nt ∈ N ( 0, σ n2,t ) is the additive Gaussian measurement noise with mean 0 and variance σ n2, t . Herein, the first and second derivatives of the angle parameters θ t are i ii denoted by θ t and θ t respectively as follows:

This work is supported by National Natural Science Foundation of China under Grant No.60472101

978-1-4244-1724-7/08/$25.00©2008IEEE

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ICALIP2008

i

i

regularization step [10] and a Metropolis-Hastings [11] based MCMC move step. So we call it improved bootstrap filter. Now we propose the following implementation of this algorithm.

ii

ωt = θ t , ωt = θ t ∼ N ( 0, σ ω2 ,t )

(2) i

Now perform Taylor series expansion to θ t and θ t , we have i

θ t ≈ θ t −T + Ts θ t −T + s

s

Ts 2 2

ii

θ t −T

s

z At time 0 Initialization Set Q = BΣ v BT , R = σ n2 FOR i = 1, … , N sample ( xi0 ) ∼ p ( x 0 ) and set

(3)

and i

i

ii

θ t ≈ θ t −T + Ts θ t −Ts

x 0 = E[ xi0 ]

(4)

s

z At time k ≥ 1 Bootstrap filtering: FOR i = 1, … , N -Draw x ik ∼ p ( x k | x ik −1 ) -Calculate

where Ts denotes a small sampling interval. Likewise, we have i

α t ≈ α t −T + Ts α t −T

s

s

(5)

Now we define the discrete-time state vector as

i

m k = p ( yk | x ik −1 )

x k = [ x1, k x2, k x3,k ]T = [θ k ωk α k ]T

(8)

END FOR - Calculate total weight:

where the superscript “T” denotes matrix transposition, and the subscript k denotes discrete time kTs . Then, according to (3)-(5), the discrete-time state dynamical model can be expressed as

N

M = ∑ mk i

(9)

i =1

x k = Ax k -1 + Bv k

FOR i = 1,… , N - Normalize:

⎡Ts / 2 0 ⎤ ⎡1 Ts 0 ⎤ (6) ⎢ ⎥ ⎢ ⎥ = 0 1 0 x k -1 + ⎢ Ts 0 ⎥ vk ⎢ ⎥ ⎢⎣ 0 Ts ⎥⎦ ⎣⎢0 0 1 ⎦⎥ where v k = [vω , k , vα , k ]T ∼ N (0, Σ v , k ) with Σ v , k =diag (σ ω2 , k ,σ α2 ,k ) and σ ω2 ,k and σ α2 ,k are the variances of ωk and α k respectively. For process noise v k , the covariance is Q k = BΣv ,k BT . 2

i

mki = M −1 m k

(10)

END FOR - Calculate the empirical covariance matrix

{xik , mki }iN=1 - Calculate D k such that Dk DTk = S k

S k of

(11) (Note that the two steps above are just preparations made for the regularization step) - Selection step Multiply/Suppress particles with high/low importance weight respectively, to obtain N random particles with importance weight 1/ N Gaussian Kernel based regularization: - Calculate the optimal bandwidth in this case [10]:

The associated observation model is then to be

yk = h(x k , nk ) = x3,k cos( x1, k ) + nk (7) 2 where nk ∈ N ( 0, σ n , k ) is the measurement noise with 2 mean 0 and covariance matrix R k = σ n , k . Note that the formulation of the NCF model is analogous to that of the nearly constant velocity model [8] used in target tracking problems.

3. Monte Carlo sampling algorithm

hopt = A ⋅ N

−

1 nx + 4

(12)

with

In this paper, the filtering problem consists of estimating online the unknown state vector x k , based on the NCF model, sequentially from the observations. To solve this problem, we use sequential Monte Carlo methods (also referred to as particle filters), which provide us a particle approximation of the distribution of the state p ( x 0:k | y1:k ) , where, e.g., x 0:k denotes [ x 0 , x1 , , x k ] . Our algorithm mainly includes three parts: a sequential importance resampling filter (also known as bootstrap filter) [9], a Gaussian kernel based

1

A = [4 /(nx + 2)]nx + 4 (13) where nx = 3 ,which is the dimension of the state

vector. FOR i = 1, … , N - Draw

ε i from the Gaussian kernel K

x ik * = x ik + hopt Dk ε i END FOR MCMC move step:

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(14)

FOR i = 1, … , N - sample a random variable u ∼ U [0,1] - calculate the acceptance probability

4. Simulation We designed 3 representative scenarios to test the improved algorithm we proposed for the problem at hand. In each test, the EKF and the bootstrap filter were executed alongside for performance comparison. A set of 100 independent Monte Carlo (MC) simulations for each scenario was executed and where possible, the Cramer-Rao lower bounds (CRLB) [12] were used to indicate the best possible performance that one can expect for a given scenario and a set of parameters. For all scenarios, the number of particles used in PFs N s = 300 . The initial frequency, phase and amplitude of the signal, were set to be 0.2π /sample, π 3 and 1 , respectively. The signal to noise ratio (SNR) set for the first scenario was 5dB whereas it was 0dB for the others. Note that, given a signal waveform in each scenario, the variance of the observation noise, i.e., σ n2, k , can be determined for a SNR level as follows σ n2, k = signal power × 10− SNR /10 . (17)

⎧ p ( yk | x ik * ) p ( x ik * | xik −1 ) ⎫ α = min ⎨1, ⎬ (15) ⎩ p ( yk | x ik ) p ( x ik | x ik −1 ) ⎭ (Note that the objective (desired) density of this MCMC step is p ( x 0:k | y1:k ) , and the derivation of (15) refers to the appendix) IF u ≤ α

let xik = xik * ELSE reject xik * . END IF END FOR Output: It is easy to compute statistical estimates from the approximation of the posterior distribution, such as:

E [ g k (x 0:k ) ] ≈

1

N

∑g N

k

( x i0:k )

(16)

i =1

In our method, the function we are interested in is marginal conditional mean of x 0:k , i.e., g k (x 0:k ) = x k .

the

4.1. The first scenario In this case, the parameters used for signal generation and filtering process were identically set as: σ ω ,k = 0.05 , σ α ,k = 10 −5 1 where k ∈ R . Results are illustrated in Fig.1.

Some remarks about the algorithm are made as follows. z The bootstrap filter has advantage that the importance weights are easily evaluated and the importance density can be easily sampled. However, as resamping is applied at every iteration, it may result in a rapid loss of diversity in particles, namely, sample impoverishment. z The regularization step replaces the dirichlet kernel with a continuous Gaussian kernel in the resampling process so as to assure that the particles are from a continuous domain. It actually improves diversity among the particles. However, after this step, the samples can not be guaranteed to asymptotically approximate those from the posterior. z The MCMC step, whose objective (desired) density is p ( x 0:k | y1:k ) , guarantees that its samples asymptotically approximate those from the posterior. To summarize, the improved bootstrap filtering method can better solve the problem of sample impoverishment. Additionally, in terms of computational complexity, it is comparable to bootstrap filter, since the only requirement is N additional generations from the Gaussian kernel K at each time step.

4.2. The second scenario In this case, the frequency used for signal generation stayed constant for a certain period, followed by a sudden jump at the 40th time step. The related parameter for the filtering algorithms was σ ω ,k = 0.06 . σ α ,k was set to be 10 −6 for both the signal generation and the filtering processes. Estimation results are shown in Fig.2.

4.3. The third scenario Similarly as in the second scenario, during the process of signal generation, the frequency stayed constant first, then it ascended with a gentle jump during the 40th and 50th time steps, finally kept constant again until the end of the observation period. The related parameter item for the filtering algorithms was σ ω , k = 0.05 . It was the same for the setting of σ α ,k as in the second scenario. We describe the estimation results in Fig.3.

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5. Conclusions

Fig. 2 (a). Estimated versus the true frequency in an example run of the MC simulation for the second scenario

The NCF model, adapted from the target tracking discipline, is also suitable for problem of frequency tracking. Based on this model, both of the PF based trackers perform consistently and have better performance than the EKF based one. Between the two PF solutions, the improved one has better performance which is closer to CRLB.

Fig. 2 (b). RMS frequency error comparison with CRLB in the second scenario

Fig. 1 (a). Estimated versus the true frequency in an example run of the MC simulation for the first scenario Fig. 3 (a). Estimated versus the true frequency in an example run of the MC simulation for the third scenario

Fig. 1 (b). RMS frequency error comparison with CRLB in the first scenario Fig. 3 (b). RMS frequency error comparison with CRLB in the third scenario

Appendix: The derivation of (15) is made as follows. The MCMC move step is based on the MetropolisHastings algorithm [11]. The key idea is that a resampled particle xik is moved to a new state xik * , only if u ≤ α , where u ∼ U [0,1] and α is the acceptance probability. Otherwise, the move is rejected.

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The objective (desired) density of the MCMC step is p ( x 0:k | y1:k ) , which can be expressed in terms of p ( x 0:k −1 | y1:k −1 ) , p ( yk | x k ) and p ( x k | x k −1 ) based on the Markov property:

p (x 0:k | y1:k ) = = =

p ( yk | x 0:k , y1:k −1 ) p (x 0:k | y1:k −1 )

[3]

H.C.So and P.C.Ching, “Adaptive algorithm for direct frequency estimation” IEE Proceedings-Radar, Sonar and Navigation, vol.151, December 2004, pp.359-364.

[4]

H.C.So, “A comparative study of three recursive least squares algorithms for single-tone frequency tracking”, signal processing, Vol,83, September 2003, pp. 20592062.

[5]

G.Li, “A stable and efficient adaptive notch filter for direct frequency estimation”, IEEE Trans. on signal processing, vol 45, August 1997, pp.2001-2009.

[6]

S.Bittani and S.M.Savaresi, “On the parameterization and desgin of an Extended Kalman filter frequency tracker”, IEEE Trans. on Automatic Control, vol.45, no. 9, 2000, pp.1718–1724.

[7]

C. Dubois, M.Davy and J.Idier, “Tracking of timefrequency components using particle filtering”, in Proc.Of IEEE International Conference on Acoustics, speech, and signal processing (ICASSP), vol.4, 2005, pp. 9-12.

[8]

X. R. Li and V. P. Jilkov, “Survey of maneuvering target tracking. Part I: Dynamic Models”, IEEE Trans. on Aerospace and Electronic Systems, vol. 39, Oct. 2003, pp. 1333–1363.

[9]

N.J. Gordon, D.J. Salmond, and A.F.M.Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation”, IEE Proc.-F, vol.140, no. 2, 1993, pp. 107113.

p ( yk | y1:k −1 )

p ( yk | x 0:k , y1:k −1 ) p (x k | x 0:k −1 , y1:k −1 ) p ( x 0:k −1 | y1:k −1 ) p ( yk | y1:k −1 ) p ( yk | x k ) p ( x k | x k −1 ) p ( yk | y1:k −1 )

p (x 0:k −1 | y1:k −1 )

(A.1)

∝ p ( yk | x k ) p (x k | x k −1 ) p (x 0:k −1 | y1:k −1 ) Consider the situations where you have generated： a) Particle xik by the resampling step, so that x i0:k = {x ik , x i0:k −1} ; b) Particle x ik * by sampling from proposal distribution q (⋅ | xik ) , so that x i0:*k = {x ik * , xi0:k −1} . So the Metropolis-Hastings acceptance probability [11] is given by:

⎧ p ( x 0:i *k | y1:k ) q (x ik | x ik * ) ⎫ α = min ⎨1, ⎬. ⎩ p ( x 0:i k | y1:k ) q (x ik * | x ik ) ⎭

(A.2)

Let q (⋅ | xik ) correspond to sampling according to (14). Since in this case q (⋅ | xik ) is symmetric in its arguments, that is q ( x ik * | x ik ) = q ( x ik | x ik * ) , the substitution of (A.1) into (A.2) yields:

⎧ p ( yk | x ik * ) p ( x ik * | xik −1 ) ⎫ α = min ⎨1, ⎬ (A.3) ⎩ p ( yk | x ik ) p ( x ik | x ik −1 ) ⎭

[10] C. Musso, N. Oudjane, and F. LeGland, Improving regularized particle filters. in sequential Monte Carlo Methods in Practice (A. Doucet, N.de Freitas, and N.J. Gordon, eds.), New York: Springer, 2001.

which is the same as equation (15).

6. References [1]

P. Stoica and R. Moses, Spectral Analysis of Signals. Upper Saddle River, NJ: Prentice-Hall, 2005.

[2]

Fulvio Gini, Monica Montanari, Lucio Verrazzani, “Maximum likelihood, ESPRIT, and periodogram frequency estimation of radar signals in K-distributed clutter”, Signal processing, vol 80, 2000, pp.11151126.

[11] C.P.Robert and G.Casella, Monte Carlo Statistical Methods. New York: Springer, 1999. [12] P.Tichavsky, C.H. Muravchik, and A.Nehorai, “Posterior Cramer-Rao bounds for discrete-time nonlinear filtering”, IEEE Trans. Signal Processing, vol.46, May 1998, pp. 1386-1396.

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