An ℓp-norm minimization approach to time delay ... - ScienceDirect

Digital Signal Processing 23 (2013) 1247–1254

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Digital Signal Processing www.elsevier.com/locate/dsp

An  p -norm minimization approach to time delay estimation in impulsive noise Wen-Jun Zeng a , H.C. So a,∗ , Abdelhak M. Zoubir b a b

Department of Electronic Engineering, City University of Hong Kong, Hong Kong Signal Processing Group, Institute of Telecommunications, Technische Universität Darmstadt, Darmstadt, Germany

a r t i c l e

i n f o

Article history: Available online 26 March 2013 Keywords: α -stable process  p -norm minimization Impulsive noise Robust estimation Time delay estimation

a b s t r a c t Estimating the time delay between two signals received at spatially separated sensors is an important topic in signal processing and has a variety of practical applications. Conventionally, time delay estimation (TDE) can be achieved in two steps. The coefficients of a finite impulse response filter used to model the subsample delay are first computed and then interpolated to produce the delay estimate. Despite its simplicity, the two-step method suffers from error accumulation, estimation bias, and is not robust to impulsive noise or outliers. To overcome these drawbacks, a family of robust algorithms for direct TDE is proposed using  p -norm minimization, with 1  p  2. Although the direct approach leads to a nonconvex optimization problem, efficient algorithms are designed for finding the global solution. Its robustness and accuracy in the presence of α -stable noise are demonstrated by comparing it with the standard two-step scheme, cross-correlator and fractional lower-order covariation method. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Time delay estimation (TDE) using two spatially separated sensors is a central problem in target localization [1–3]. From the delay estimates, nonlinear equations can be constructed to solve for the source position [4–6]. Apart from positioning, TDE is useful in other important applications such as synchronization in wireless communications [7], echo cancellation, speech enhancement and multimedia [8,9]. Generalized cross-correlation (GCC) [1] and parameter estimation [10,11] are two conventional approaches for TDE. For GCC, each received signal is passed through a prefilter prior to crosscorrelation. Basically, the prefilters are employed to enhance the frequency bands where the signal-to-noise ratio (SNR) is high and to attenuate the bands with low SNR. However, their implementation requires knowledge of the signal and noise spectra, which are usually unknown, and thus have to be estimated from observations. Due to inaccuracies associated with spectral estimation from finite data length, the ideal GCC performance is difficult to achieve in practice. On the other hand, the parameter estimation approach utilizes a finite impulse response (FIR) filter [10–12] to model the delay and consists of the following two steps. First, the coefficients of the FIR filter are computed according to the least squares (LS) criterion. The tap coefficients are then interpolated to produce the delay estimate. Despite its simplicity, this two-step TDE method

*

Corresponding author. E-mail address: [email protected] (H.C. So).

1051-2004/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.dsp.2013.03.013

suffers from error accumulation, that is, the estimation error of the FIR filter coefficients will affect the TDE accuracy, leading to a bias. To reduce the delay estimation bias, the filter length needs to be increased [11]. A common deficiency of these two approaches is that they generally assume the disturbances in the sensor outputs to be Gaussian, although the noise components in practice often exhibit non-Gaussian properties. One important class of nonGaussian noise that is frequently encountered in many practical systems is impulsive noise. Among different statistical representations, α -stable noise [13–15] is an important and widely used impulsive noise model. Since second-order and higher-order statistics of the α -stable distribution are infinite, the GCC approach which relies on second-order statistics will give poor performance in the presence of impulsive noise. Similarly, the parameter estimation methodology based on the LS criterion also performs unreliably because the LS estimator is not robust against α -stable noise. To improve robustness, several TDE methods have been developed. In [16], the least  p -norm is suggested in the two-step TDE method. Since it has been shown [15] that robustness to outliers can be enhanced by using a smaller value of p < 2, the approach in [16] is more robust to impulsive noise compared to its standard counterparts in [10,11]. However, it is still a two-step method and hence it suffers from error accumulation and results in estimation bias for short FIR filter lengths. More recently, Liu et al. [17] have proposed a robust TDE method but it is solely designed for cyclostationary source signals. Apart from TDE, other applications which make use of least  p -norm include [18,19]. In this work, we devise an  p -norm approach for TDE in a direct and effective manner for general random signals embedded in impulsive noise.

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The rest of this paper is organized as follows. Section 2 formulates the TDE problem and provides a brief review of the two-step TDE approach. In Section 3, we present in detail the least  p norm algorithm for direct TDE. Section 4 describes the probabilistic model of the α -stable process and discusses selection of p. Numerical examples are included to demonstrate the robustness of the proposed algorithm in Section 5. Finally, conclusions are drawn in Section 6. The notation used in this paper is introduced. Bold upper-case and lower-case letters denote matrices and vectors, respectively. The superscript (·) T , | · | and  ·  p represent transpose, absolute value and  p -norm of a vector, respectively, with p = 2 being the Euclidean norm. The operator E{·} denotes expectation,  denotes the element-wise multiplication and I is the identity matrix. Finally, R and Z+ are used to represent the sets of real-valued and positive integers, respectively. 2. Problem statement and background 2.1. Signal model

h



M 

x2 [n] −

n = M +1

2 h i x1 [n − i ]

i =− M

−1

= R 11  d12

(5)

where

 h = [ h−M , h − M +1 , . . . ,  h M ]T

(6)

is the estimate of the coefficient vector,

⎡   R x1 x1 (0) R x1 x1 (−1) ⎢   R ( 1 ) R x1 x1 (0) x x 1 1 ⎢  R 11 = ⎢ .. . .. ⎣ .  R x1 x1 (2M )  R x1 x1 (2M − 1)

⎤ ···  R x1 x1 (−2M ) ···  R x1 x1 (1 − 2M ) ⎥ ⎥ ⎥ (7) .. .. ⎦ . .  ··· R x1 x1 (0)

T  d12 = R x1 x2 (0), . . . , R x1 x2 (2M )

x2 [n] = β s[n − D ] + v 2 [n],

n = 1, 2, . . . , N

 h = arg min

N −M

is the estimated (2M + (2M + 1) auto-correlation matrix of 1N)−× M x1 [n] with  R x1 x1 (m) = n= M +1 x1 [n]x1 [n − m], and

The observed discrete-time signals at the two sensors are

x1 [n] = s[n] + v 1 [n],

Based on (3), a noncausal FIR filter with coefficients {h i }iM=− M is employed to model the time shift of D for x1 [n]. This TDE procedure consists of two steps as follows. First, the filter coefficients are estimated according to the LS criterion:

(1)

where s[n] is the unknown random source signal, β and D are the attenuation factor and time delay between the sensors, respectively, v 1 [n] and v 2 [n] are uncorrelated impulsive additive noises and they are independent of s[n]. Note that the time delay is not restricted to be an integer multiple of the sampling period and hence generally D ∈ R. Although the attenuation factor is assumed to be β = 1 in [10,11], this holds only when both sensors are in the far field of the source and have similar received signal strengths. However, when the signal strengths at the two sensors differ, this assumption does not hold and we must introduce β in the signal model. The task of passive TDE is to find the delay D based on the N discrete observations {x1 [n], x2 [n]}nN=1 .

(8)

is the estimated N −Mcross-correlation vector of x1M[n] and x2 [n] with  n= M +1 x1 [n]x2 [n − m]. Once {h i }i =− M are obtained, the second step of TDE is to use the sinc interpolation [11] to compute the delay estimate:

 R x1 x2 (m) =

 D = arg max h( D ),  h( D ) = D

M 

 h i sinc( D − i ).

(9)

i =− M

Clearly, the two-step scheme needs searching for the peak of the h( D ). A grid search can be used to find interpolated delay profile  the peak and the grid size determines the precision. This two-step procedure has been widely used [10–12] because the closed-form expression of the FIR filter coefficient vector can be easily obtained by (5). Despite its simplicity, this approach has three intrinsic limitations:

2.2. Review of two-step TDE method Chan et al. [10,11] have proposed to use an FIR filter to estimate the fractional time delay. The underlying idea is that according to the convolution theorem, s[n − D ] can be expressed as

s[n − D ] =

∞ 

s[n − i ] sinc(i − D )

1) It suffers from error accumulation, that is, the estimation error of the FIR filter coefficients will affect the TDE accuracy. 2) The sinc interpolation based two-step TDE estimator is biased for a finite M. To decrease the delay estimation bias, the filter length needs to increase [11]. 3) The LS criterion is optimal in the presence of Gaussian noise, but it is not robust in the non-Gaussian impulsive noise case.

i =−∞

≈

M 

s[n − i ] sinc(i − D )

(2)

i =− M

where sinc(t ) = sin(π t )/(π t ) is the sinc function and the approximation error decreases with an increasing M ∈ Z+ . Note that without loss of generality, the Nyquist rate sampling is assumed. From (1) and (2), we obtain

x2 [n] ≈ β

M 

x1 [n − i ] sinc(i − D ) + v [n]

(3)

i =− M

where v [n] is the combined noise component, which is

v [n] = v 2 [n] − β

M  i =− M

v 1 [n − i ] sinc(i − D ).

(4)

To overcome the third drawback, Ma and Nikias [16] have proposed to use the least  p -norm criterion:

 h = arg min h

N −M n = M +1

p  M      h i x1 [n − i ] x2 [n] −  

(10)

i =− M

with 1 < p < 2 to replace the LS criterion in [16], which is referred to as the fractional lower-order covariation (FLOC) method. The least  p -norm approach is more robust to impulsive noise if p < 2. The gradient descent method is suggested in [16] to solve for (10), but there is no discussion on the selection of the step size parameter. In fact, the convergence rate of the gradient descent method is quite slow for solving the  p -norm minimization problem. Moreover, the gradient descent method proposed in [16] cannot be applied to the case of p = 1 because the objective function is not differentiable for p = 1. Nevertheless, (10) is still a two-step procedure and also suffers from the first two drawbacks.

W.-J. Zeng et al. / Digital Signal Processing 23 (2013) 1247–1254

In this paper, we propose a novel approach for direct TDE in the presence of impulsive noise. The time delay is estimated explicitly by minimizing an  p -norm based objective function. Although the cost function is nonconvex with respect to the delay, we develop an efficient optimization algorithm with moderate complexity to obtain the global solution.

Although f p ( D , β) is a two-dimensional nonconvex function, it is convex with respect to β for a fixed D. Hence the global optimality of β for a given D can be guaranteed. In the first step of our optimization algorithm, we aim at finding the global optimum β for a given D, denoted by β ∗ ( D ). This problem is classified into three cases, according to the value of p. Case 1: p = 2. This is the simplest case because

3. Direct robust TDE algorithm

3.1. Objective function for TDE Combining (3) and (5) and then generalizing the expression which is analogous to (10), we propose to estimate D and β directly by minimizing the  p -norm cost function:

p  N −M  M       { D , β } = min x1 [n − i ] sinc(i − D ) x2 [n] − β   D ,β

(11)

i =− M

where 1  p  2. Note that when both v 1 [n] and v 2 [n] are Gaussian processes, the combined noise v [n] of (4) is also Gaussian distributed. In this case, we should choose p = 2 for accurate delay estimation, while 1  p < 2 is employed when v 1 [n] and v 2 [n] model impulsive noise. 3.2. Efficient algorithm for nonconvex minimization The two-dimensional nonlinear minimization problem of (11) is challenging because the objective function is nonconvex with respect to D and hence contains local minima. The conventional gradient descent and Newton methods easily result in local minimization when the initial estimates are not sufficiently close to the global solution. Therefore it is not appropriate to apply the standard numerical methods to solve for (11). In this section, we design an efficient optimization algorithm to find the global minimizer of (11) for 1  p  2. We first write a compact matrix formulation of (11). By defining the following two vectors 

T

x2 = x2 [ M + 1], x2 [ M + 2], . . . , x2 [ N − M ]

∈ R N −2M

T c ( D ) = sinc(− M − D ), . . . , sinc( M − D ) ∈ R2M +1 

(12) (13)

and the ( N − 2M ) × (2M + 1) Toeplitz matrix

⎡

x1 [2M + 1] ⎢  ⎢ x1 [2M + 2]

X1 = ⎣

x1 [2M ] x1 [2M + 1]

.. . x1 [ N ]

.. .

x1 [ N − 1]

2



In this section, the  p -norm minimization approach for robust TDE is devised as follows.

n = M +1

1249

··· ··· .. .

x1 [1] x1 [2]

⎤

⎥ ⎥ .. ⎦ . · · · x1 [ N − 2M ]

(14)

f 2 ( D , β) = β b( D ) − x2 

(18)

is a quadratic function of β for a fixed D. Performing

 ∂ f 2 ( D , β)   ∂β

β=β ∗

=0

(19)

leads to

β∗ =

b T ( D ) x2

(20)

b( D )2

whose complexity is O ( N ). Substituting (20) into (18) and after manipulations, the minimization problem with β being eliminated is reduced to

Q 2 ( D ) = x2 2 −

min Q 2 ( D ), D

| b T ( D ) x 2 |2 b( D )2

(21)

with only a single variable D to be optimized. We refer this onedimensional function Q 2 ( D ) to the delay profile. Similar to the second step of the two-step TDE method, we can use a grid search to find the minimum of the delay profile and compute the delay D = arg min D Q 2 ( D ). To be more specific, we first deestimate by  termine a search range [ D min , D max ] based on its admissible values and a search precision  D, which corresponds to delay resolution, then compute the delay profile over the search grids and find the D is obtained, the attenuation minimum. Once the delay estimate  D into (20): factor estimate is calculated by substituting 

= β

b T ( D ) x2

b (  D )2

(22)

.

The delay estimation algorithm for p = 2 is summarized in Algorithm 1. Algorithm 1 Time delay estimation algorithm for p = 2. Input: The received signals {x1 [n], x2 [n]}nN=1 at the two sensors, the delay search parameters D min , D max , and  D, and the approximation order M. . Output: Time delay estimate  D and attenuation factor estimate β Algorithm: for D = D min :  D : D max do Compute b( D ) = X 1 c ( D ) with c ( D ) and X 1 defined in (13) and (14), respectively. Compute Q 2 ( D ) according to (21). end for Plot the delay profile and find its minimum  D = arg min D Q 2 ( D ).  using (22). Compute the attenuation factor estimate β

the minimization problem of (11) can be rewritten as



p

min f p ( D , β) = minβ X 1 c ( D ) − x2  p D ,β

(15)

D ,β



where the  p -norm x p of a vector x = [x1 , . . . , x N ] T is defined as

  x p =

N 

1 / p | xi |

p

(16)

.

i =1



For the purpose of notation simplicity, we define b( D ) = X 1 c ( D ), and the objective function is now expressed as

 p f p ( D , β) = β b( D ) − x2  . p

Case 2: p = 1. This corresponds to the least absolute deviation problem

(17)



min f 1 ( D , β) = minβ b( D ) − x2 1 β

β

(23)

for a fixed D. Although there is no closed-form expression for β ∗ from (23), it can easily be computed by the weighted median of the two sequences x2 = [x2 [ M + 1], x2 [ M + 2], . . . , x2 [ N − M ]] T and b( D ) = [b M +1 ( D ), b M +2 ( D ), . . . , b N − M ( D )] T . Eq. (23) is equivalent to

min f 1 (β) = min β

β

N −M n = M +1

     bn ( D )β − x2 [n]   bn ( D ) 

(24)

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W.-J. Zeng et al. / Digital Signal Processing 23 (2013) 1247–1254

where |bn ( D )|  0 is the nonnegative weight. Defining a new sequence

y [n] =

x2 [n] bn ( D )

,

n = M + 1, . . . , N − M

   β ∗ = WMED bn ( D ), y [n]

(26)

to represent this weighted medium. The computation procedure is outlined in Algorithm 2. The major operation in the weighted median computation is to sort the weight coefficients. Hence the computational complexity for solving (23) is O ( N log N ), assuming that a quick sorting algorithm is employed.

Input: The weight coefficients {|bn ( D )|}nN=−MM+1 and the data sequence { y [n]}nN=−MM+1 . Output: The weighted median β ∗ = WMED(|bn ( D )|, y [n]). n= M +1

|bn ( D )|.

2. Sort the data sequence { y [n]}nN=−MM+1 in ascending order with the corresponding

concomitant weights {|bn ( D )|}nN=−MM+1 . 3. Sum the concomitant weights, beginning with |b M +1 ( D )| and increasing the order. 4. The weighted median β ∗ is y [m] whose weight leads to the inequality m n= M +1 |bn ( D )|  b 0 to hold first.

Case 3: 1 < p < 2. In this case, there is also no closed-form exp pression for β ∗ in minimizing f p (β) = β b( D ) − x2  p for a fixed D. We will develop a simple fixed-point iteration method to find the optimal point. By defining the residual vector

r = β b( D ) − x2 = [r M +1 , . . . , r N − M ] T

(27)

the objective function f p (β) can be expressed as p

f p (β) = f p (r ) = r  p =

N −M

|rn | p .

(28)

n = M +1

It is not difficult to verify that the partial derivative with respect to rn is

∂ fp = p |rn | p −2 rn , ∂ rn

n = M + 1, . . . , N − M

(29)

and hence the gradient of f p (r ) with respect to r is

  ∂ fp ∂ fp ∂ fp T = ,..., = p | r | p −2  r ∂r ∂ r M +1 ∂ r N −M

(30)

where |r | p −2 = [|r M +1 | p −2 , . . . , |r N − M | p −2 ] T . Defining a diagonal matrix



W (β) = diag |r M +1 | p −2 , . . . , |r N − M | p −2



(31)

(30) can be rewritten as

∂ fp = pW (β)r . ∂r

(32)

Clearly W (β) is positive definite. The ( N − 2M ) × 1 Jacobian matrix of r (β) is given by

∂r = b( D ). ∂β

(33)

Then the first-order derivative of f p (β) with respect to β is computed as

∂r ∂β

T

∂ fp ∂r

  = pb T ( D ) W (β) β b( D ) − x2 .

(34)

The optimal point satisfies ∇ f p (β) = 0, which leads to

β=

b T ( D ) W (β)x2 b T ( D ) W (β)b( D )

(35)

.

When p = 2, we have W (β) = I . Hence (35) reduces to (20) for p = 2. Introduce another vector whose elements are equal to the square root of the diagonal elements of W (β):

T

φ(β) = |r M +1 |( p −2)/2 , . . . , |r N − M |( p −2)/2 .

(36)

Eq. (35) can be rewritten as

β=

Algorithm 2 Computation of weighted median.

1. Determine the threshold b0 = (1/2)

∇ f p (β) =

(25)

the optimal β ∗ is the weighted median of the sequence { y [n]}nN=−MM+1 with the corresponding weights {|bn ( D )|}nN=−MM+1 . We use the notation

N −M



(φ(β)  b( D ))T (φ(β)  x2 ) . φ(β)  b( D )2

(37)

Eq. (37) inspires us to use the following fixed-point iteration to find the optimal β ∗ ( D )

(k+1) = β

(k) )  b( D ))T (φ(β (k) )  x2 ) (φ(β , ( k )  )  b( D )2 φ(β

k = 0, 1, . . .

(38)

(k) denotes the estimate of β in the kth iteration. The iniwhere β (0) can be obtained using the LS solution of (20). It tial value β is worth noting that the fixed-point iteration of (38) is equivalent to the well-known iteratively reweighted least squares (IRLS) algorithm [15,20–22] whose convergence has been proved in [20]. Since only vector inner products need to compute, the complexity is O ( N ) in each iteration. The total complexity of the IRLS algorithm is thus O ( N IRLS N ) where N IRLS is the number of iterations required for convergence. Typically, N IRLS is a value of several tens [23]. Therefore, the complexity for solving the optimal β is moderate. After obtaining the optimal β ∗ ( D ) and substituting it back into the objective function, gives the delay profile



p

Q p ( D ) =  β ∗ ( D ) b ( D ) − x2  p .

(39)

The direct TDE algorithm based on  p -norm minimization with 1  p < 2 is summarized in Algorithm 3. Algorithm 3 Time delay estimation algorithm for 1  p < 2. Input: The received signals {x1 [n], x2 [n]}nN=1 at the two sensors, the delay search parameters D min , D max , and  D, and the approximation order M. . Output: Time delay estimate  D and attenuation factor estimate β Algorithm: for D = D min :  D : D max do Compute b( D ) = X 1 c ( D ) with c ( D ) and X 1 defined in (13) and (14). if p = 1 then Compute the optimal β ∗ ( D ) by weighted median according to Algorithm 2. else Compute the optimal β ∗ ( D ) using the fixed-point iteration of (38). end if Compute the delay profile Q p ( D ) using (39). end for Plot the delay profile and find its minimum  D = arg min D Q p ( D ) and the corre = β ∗ ( sponding attenuation factor estimate β D ).

Denoting the number of grid points for D as N D = ( D max − D min )/ D and recalling the complexity for determining β , the computational load of the proposed TDE algorithm is summarized as follows, O ( N D N ), O ( N D N log N ) and O ( N D N N IRLS ) for p = 2, p = 1 and 1 < p < 2, respectively. As a final remark, the proposed approach reduces the two-dimensional optimization problem to a one-dimensional delay profile minimization, where a grid search

W.-J. Zeng et al. / Digital Signal Processing 23 (2013) 1247–1254

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is used to search for the global minimum. Clearly, the estimate precision is dominated by the grid size. It is not advisable to use a very fine grid uniformly because this will greatly increase the computational complexity. In order to achieve high precision and reduce complexity, a multi-resolution grid refinement is adopted. Specifically, we first obtain an approximate region where the delay appears using a coarse grid search, then use a finer grid around this region. The above refinement process is repeated until a satisfactory precision is obtained. 4. Noise model and selection of p The probability density function (PDF) of impulsive noise exhibits heavier tails than those of the Gaussian distribution. The α -stable process is widely used to model impulsive noise [13,16]. Here, we adopt the symmetric α -stable (Sα S) distribution with zero-location as the impulsive noise model, whose characteristic function is expressed as



ϕ (ω) = exp −γ α |ω|α



(40)

where 0 < α  2 is called the characteristic exponent that describes the tail of the distribution, and γ > 0 is the scale. The smaller the value of α , the more impulsive the noise is. To quantify the relative strength between signal and noise, a generalized signal-to-noise ratio (GSNR)

GSNR =

E{s2 [n]}

γα

(41)

is adopted. In our study, it is assumed that the noise processes of the two sensors v 1 [n] and v 2 [n] satisfy the same α -stable distribution. When 1 < α < 2, there are no closed-form expressions for the PDFs of v 1 [n] and v 2 [n] and so does the combined noise v [n]. In addition, the PDF of v [n] will become more complicated than that of v 1 [n] or v 2 [n] because it contains the unknown parameters D and β . Due to the difficulty in expressing the PDF of v [n], it is quite difficult to obtain the maximum likelihood (ML) estimates of D and β . This is still an open problem that has not been addressed in parameter estimation in the presence of α -stable noise. The value of p is crucial in the proposed TDE algorithm. Clearly, the optimal p is related to α . When α = 2, v 1 [n] and v 2 [n] are Gaussian and v [n] is also Gaussian, and thus the optimal p is α = 2. Since the ML function is generally not available, it is difficult to analytically derive the optimal p for 1  α < 2, which remains an open problem. Nevertheless, we find in Section 5 that 1  p  α is an appropriate choice for achieving robustness. 5. Numerical results In this section, numerical simulations are conducted to demonstrate the performance of the robust TDE approach in additive impulsive noise. The source signal s[n] is a zero-mean white Gaussian random sequence. The true time delay and attenuation factor are set to D = 2.76 and β = 0.95. The approximation order parameter is M = 10. The delay search range is [ D min , D max ] = [−10, 10]. In order to reduce computational complexity and achieve a high search precision, a multi-resolution grid refinement strategy is adopted. Four-level resolutions are used with search step sizes of  D = 10−2 , 10−4 , 10−6 , and 10−8 . That is, the final attainable search precision is 10−8 . In the first search level, the delay profile needs to compute at N D = ( D max − D min )/10−2 = 2000 points. In each of the remaining three resolution levels, we only need to compute the delay profile at N D = 1/10−2 = 100 points. Monte Carlo trials are carried out to evaluate the performance of the TDE algorithms. Two statistical performance measures are used. We adopt the probability of success as the first performance

Fig. 1. Delay profiles of different TDE methods in Gaussian noise with SNR = 0 dB. (a) Cross-correlation; (b) Proposed algorithm with p = 2; (c) Proposed algorithm with p = 1.3; (d) Two-step approach with sinc interpolation. The red vertical line represents the true delay D = 2.76.

index to evaluate robustness to outliers. In the presence of impulsive noise, the conventional TDE methods are not robust and will give incorrect estimates. A trial is considered to be failed if the delay error is larger than 0.5. This aligns with synchronization of wireless communication systems where a delay estimation precision within 0.5 sampling interval is required [7]. Therefore, we take the threshold of 0.5 as the indication of small error. The second index is the root mean square error (RMSE):

  Mc  1  RMSE =  ( D m − D )2 Mc

(42)

m =1

where M c is the number of Monte Carlo trials and  D m is the delay estimate of the mth run. 5.1. Demonstration of robustness In the first simulation, the robustness of the proposed TDE, cross-correlation, and two-step TDE methods is compared. The signal length is N = 200. We test these TDE methods in both Gaussian and Sα S noise processes. Fig. 1 plots the delay profiles of the cross-correlator, the proposed algorithm with p = 2 and p = 1.3, and the two-step approach, for Gaussian noise with SNR = 0 dB. From the minimum or maximum of the delay profile, we see that all four methods provide accurate delay estimation in Gaussian noise. Fig. 2 shows the delay profiles in Sα S noise with α = 1.4 and GSNR = 0 dB. We see that the proposed TDE algorithm with p = 1.3 is robust to both Gaussian and impulsive noise. On the other hand, the cross-correlation method, which is based on second-order moments, performs unreliably in α -stable noise and its estimate is totally wrong because the noise has infinite second-order moments. The direct 2 -norm minimization and twostep sinc interpolation algorithms also have large estimation biases and they are inferior to the proposed scheme with p = 1.3. Figs. 1 and 2 also show that there are many local minima in the objective functions. Hence the global convergence cannot be guaranteed if gradient or Newton method is employed. Nevertheless, our proposed TDE algorithm using a one-dimensional grid search with moderate complexity is preferable since it is able to find the global solution, particularly for a sufficiently smooth objective function.

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Fig. 4. RMSE versus p with

α = 1.4.

Fig. 2. Delay profiles of different TDE methods in α -stable noise with α = 1.4 and GSNR = 3 dB. (a) Cross-correlation; (b) Proposed algorithm with p = 2; (c) Proposed algorithm with p = 1.3; (d) Two-step approach with sinc interpolation. The red vertical line represents the true delay D = 2.76.

Fig. 5. Probability of success versus N.

Fig. 3. Probability of success versus p.

justification is that the minimum dispersion criterion for α -stable noise is similar to the minimum variance criterion for Gaussian noise. Therefore, as a rule of thumb, p should be chosen close to α to attain the minimum RMSE in α -stable noise.

5.2. Selection of p

5.3. Statistical performance comparison

As pointed out in Section 4, p is an important parameter for the robust TDE approach. In this test, we investigate the performance versus p when the characteristic exponent of the α -stable noise is fixed to α = 1.4. The GSNR is 6 dB and the data length is N = 200. Figs. 3 and 4 plot the probability of success and RMSE versus p, respectively. Here, p is varied from 1 to 2 with a step of 0.05 and at each value of p, 2000 Monte Carlo trials are performed. We see that the estimator with a smaller p achieves a higher probability of success, and thus is more robust to outliers. This aligns with the conclusion that the lower-order moments are more outlier-resistant. However, minimum RMSE is not attained by the smallest value of p. In other words, a small p is more outlier-resistant but does not necessarily achieve higher estimation accuracy. This is a general behavior of robust techniques where robustness is achieved at the cost of statistical efficiency. The p with the highest estimation accuracy is around α = 1.4. An intuitive

The statistical performance of the proposed TDE algorithm and FLOC method [16] with different values of p, as well as the twostep sinc interpolation method is compared. We study their probabilities of success and RMSEs versus N, GSNR and α . Figs. 5 and 6 show the results versus N ∈ [50, 500] at α = 1.4 and GSNR = 6 dB. We see that the proposed estimator provides the largest success probability with p = 1 and the smallest RMSE with p = 1.3 for all data lengths in Figs. 5 and 6, respectively. While the methods with p = 2 perform poorly in both plots. Furthermore, Fig. 6 indicates the error accumulation problem in [16] which is also a two-step solution, because of its inferiority over the direct approach. The results versus GSNR ∈ [0, 12] dB at α = 1.4 and N = 200 are plotted in Figs. 7 and 8, which generally align with our observations in Figs. 5 and 6. Similar findings are also obtained in Figs. 9 and 10, which show the success probability and RMSE performance versus α ∈ [1.1, 2], respectively, at GSNR = 5 dB and N = 200.

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Fig. 6. RMSE versus N.

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Fig. 9. Probability of success versus

Fig. 10. RMSE versus

Fig. 7. Probability of success versus GSNR.

α.

α.

In summary, the conventional LS criterion for TDE is not robust in the presence of impulsive noise. On the other hand, the FLOC approach and  p -norm minimization with p < 2 leads to an improved performance particularly for low GSNR, short signal length, and/or small α . Moreover, higher outlier-resistance is attained for smaller p while the optimal p achieving minimum RMSE is close to α . It is also seen that the proposed algorithm is generally superior to the FLOC method. 6. Conclusion

Fig. 8. RMSE versus GSNR.

We have proposed a class of TDE algorithms for impulsive noise using the  p -norm minimization. Unlike the conventional two-step TDE method which interpolates the FIR filter coefficients, our approach directly estimates the time delay. An efficient algorithm that can lead to global convergence with moderate complexity has been developed for solving the resultant nonconvex optimization problem. Simulation results under α -stable noise demonstrate that the direct method is more robust against impulsive noise and outperforms the conventional two-step TDE method in terms of outlier-resistance and estimation accuracy. Although we have given

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a rule of thumb on selection of p in α -stable noise, determining the optimal p still remains an open problem and it will be an interesting research direction. The PDF of α -stable noise has no closed-form expression and the TDE problem involves two α -stable noise sequences received from two sensors, result in great challenges to determine the optimal p, even when we know the statistical properties of impulsive noise. References [1] C.H. Knapp, G.C. Carter, The generalized correlation method for estimation of time delay, IEEE Trans. Acoust. Speech Signal Process. 24 (4) (1976) 320–327. [2] A.H. Quazi, An overview on the time delay estimate in active and passive systems for target localization, IEEE Trans. Acoust. Speech Signal Process. 29 (3) (1981) 527–533. [3] G.C. Carter, Coherence and time delay estimation, Proc. IEEE 75 (2) (1987) 236– 255. [4] H.C. So, Source localization: Algorithms and analysis, in: S.A. Zekavat, M. Buehrer (Eds.), Handbook of Position Location: Theory, Practice and Advances, Wiley–IEEE Press, 2011. [5] K.W.K. Lui, F.K.W. Chan, H.C. So, Accurate time delay estimation based passive localization, Signal Process. 89 (9) (2009) 1835–1838. [6] A. Yeredor, E. Angel, Joint TDOA and FDOA estimation: A conditional bound and its use for optimally weighted localization, IEEE Trans. Signal Process. 59 (4) (2011) 1612–1623. [7] E.G. Strom, S. Parkvd, S.L. Miller, B.J. Ottersten, Propagation delay estimation in asynchronous direct-sequence code-division multiple access systems, IEEE Trans. Commun. 44 (1) (1996) 84–93. [8] J. Benesty, J. Chen, Y. Huang, Microphone Array Signal Processing, SpringerVerlag, Berlin, Germany, 2008. [9] P.G. Georgiou, P. Tsakalides, C. Kyriakakis, Alpha-stable modeling of noise and robust time-delay estimation in the presence of impulsive noise, IEEE Trans. Multimedia 1 (3) (1999) 291–301. [10] Y.T. Chan, J.M. Riley, J.B. Plant, A parameter estimation approach to time delay estimation and signal detection, IEEE Trans. Acoust. Speech Signal Process. 28 (1) (1980) 8–15. [11] Y.T. Chan, J.M. Riley, J.B. Plant, Modeling of time delay and its application to estimation of nonstationary delays, IEEE Trans. Acoust. Speech Signal Process. 29 (3) (1981) 577–581. [12] H.C. So, On time delay estimation using an FIR filter, Signal Process. 81 (8) (2001) 1777–1782. [13] C.L. Nikias, M. Shao, Signal Processing with Alpha-Stable Distributions and Applications, John Wiley & Sons, New York, USA, 1995. [14] B. Weng, K.E. Barner, TR-MUSIC-A robust frequency estimation method in impulsive noise, Signal Process. 86 (1) (2006) 1477–1487. [15] E.E. Kuruoglu, Signal processing in α -stable noise environments: A least l p norm approach, Ph.D. thesis, University of Cambridge, UK, 1998. [16] X. Ma, C.L. Nikias, Joint estimation of time delay and frequency delay in impulsive noise using fractional lower order statistics, IEEE Trans. Signal Process. 44 (11) (1996) 2669–2687. [17] Y. Liu, T. Qiu, H. Sheng, Time-difference-of-arrival estimation algorithms for cyclostationary signals in impulsive noise, Signal Process. 92 (9) (2012) 2238– 2247. [18] F. Moghimi, A. Nasri, R. Schober, L p -norm spectrum sensing for cognitive radio networks impaired by non-Gaussian noise, in: Proc. GLOBECOM, 2009, pp. 1–6. [19] A. Navia-Vazquez, J. Arenas-Garcia, Combination of recursive least p-norm algorithms for robust adaptive filtering in alpha-stable noise, IEEE Trans. Signal Process. 60 (3) (2012) 1478–1482. [20] R.H. Byrd, D.A. Pyne, Convergence of the iteratively reweighted least squares algorithm for robust regression, Technical Report 313, John Hopkins University, 1979. [21] E.E. Kuruoglu, P.J.W. Rayner, W.J. Fitzgerald, Least l p -norm impulsive noise cancellation with polynomial filters, Signal Process. 69 (1) (1998) 1–14. [22] J. Schroeder, R. Yarlagadda, J. Hershey, L p normed minimization with applications to linear predictive modeling for sinusoidal frequency estimation, Signal Process. 24 (2) (1991) 193–216. [23] I. Daubechies, R. DeVore, M. Fornasier, C.S. Gunturk, Iteratively re-weighted least squares minimization for sparse recovery, Commun. Pure Appl. Math. 63 (1) (2010) 1–38.

Wen-Jun Zeng received the M.S. degree in electrical engineering from Tsinghua University, Beijing, China, in 2008. From 2006 to 2009, he was a Research Assistant with Tsinghua University. From 2009 to 2011, he was a faculty member with the Department of Communication Engineering, Xiamen University, China. He is now a Senior Research Associate with Department of Electronic Engineering, City University of Hong Kong. His research interests lie in the areas of signal processing and computational mathematics, including convex optimization, array processing, sparse approximation, channel identification, deconvolution and inverse problem, with applications to wireless radio and underwater acoustic communications.

Hing Cheung So was born in Hong Kong. He obtained the B.Eng. degree from the City University of Hong Kong and the Ph.D. degree from The Chinese University of Hong Kong, both in Electronic Engineering, in 1990 and 1995, respectively. From 1990 to 1991, he was an Electronic Engineer at the Research and Development Division of Everex Systems Engineering Ltd., Hong Kong. During 1995–1996, he worked as a Post-Doctoral Fellow at The Chinese University of Hong Kong. From 1996 to 1999, he was a Research Assistant Professor at the Department of Electronic Engineering, City University of Hong Kong, where he is currently an Associate Professor. His research interests include statistical signal processing, fast and adaptive algorithms, signal detection, parameter estimation, and source localization. He has been on the editorial boards of IEEE Transactions on Signal Processing, Signal Processing, Digital Signal Processing and ISRN Applied Mathematics as well as a member in Signal Processing Theory and Methods Technical Committee of the IEEE Signal Processing Society.

Abdelhak M. Zoubir received the Dr.-Ing. from Ruhr-Universität Bochum, Germany, in 1992. He was with Queensland University of Technology, Australia, from 1992 to 1998, where he was an Associate Professor. In 1999, he joined Curtin University of Technology, Australia, as a Professor of Telecommunications and was Interim Head of the School of Electrical and Computer Engineering from 2001 to 2003. In 2003, he moved to Technische Universität Darmstadt, Germany, as Professor of Signal Processing and Head of the Signal Processing Group. His research interest lies in statistical methods for signal processing with emphasis on bootstrap techniques, robust detection and estimation and array processing applied to telecommunications, radar, sonar, car engine monitoring, and biomedicine. He published more than 300 journal and conference papers on these areas. Professor Zoubir was Technical Chair of the 11th IEEE Workshop on Statistical Signal Processing (SSP 2001), General Co-Chair of the 3rd IEEE International Symposium on Signal Processing and Information Technology (ISSPIT 2003), and of the 5th IEEE Workshop on Sensor Array and Multi-channel Signal Processing (SAM 2008). He is the General Co-Chair of SPAWC 2013, to be held in Darmstadt, EUSIPCO 2013 to be held in Marrakesh, Morocco, and Technical Co-Chair of ICASSP-14 to be held in Florence, Italy. He is an IEEE Distinguished Lecturer (Class 2010–2011). He was an Associate Editor of the IEEE Transactions on Signal Processing (1999–2005), a Member of the Senior Editorial Board of the IEEE Journal on Selected Topics in Signal Processing (2009–2011), and currently serves on the Editorial Boards of the European Association of Signal Processing (EURASIP) journals Signal Processing and the Journal on Advances in Signal Processing (JASP). He is the Editor-in-Chief of the IEEE Signal Processing Magazine (2012–2014). He is Past-Chair (2012) of the IEEE SPS Technical Committee Signal Processing Theory and Methods (SPTM) [Chair (2010–2011), Vice-Chair (2008–2009), and Member (2002–2007)], and a Member of the IEEE SPS Technical Committee Sensor Array and Multi-channel Signal Processing (SAM) (2007–2012). He also serves on the Board of Directors of the EURASIP.