A new method for unbiased time‐delay estimation in noisy environments

Abstract

When input data are contaminated with noise, the least mean square time-delay estimation (LMSTDE) actually gives a biased estimate. In this paper, we propose and analyse a new adaptive filter structure for time-delay estimation (TDE), which can eliminate this bias. A new adaptive criterion is then constructed. We determine the corresponding analytical solution, and develop the stochastic gradient algorithm to calculate the optimum solution. Convergence of the stochastic gradient algorithm is established, and upper bounds on the step sizes are deduced for guaranteed convergence. Simulation results are included to illustrate the superiority of the new model and corroborate the theoretical developments.

Complex analytic signals applied on time delay estimation

“…If the condition for the perfect coherence is not satisfied, i.e., if ⎪γ s 1 s 2 (ω)⎪ ≠ 1, then the function SNR TD (ω) characterizes the estimation performance. As shown in [24], the cross-correlation function of the narrow-band signal has a number of neighboring peaks (local maximums) with approximately the same magnitude (quasi-periodic function). In case of small SNR, the reliability of cross-correlation peak determination (corresponding to the true value of differential delay) decreases.…”

“…A lot of algorithms presented to determine differential delay are based on the cross-correlation function of sensor signals. A method to determine differential delay based on FIR coefficient computation of one of the signals is presented in [24]. FIR filter coefficients are determined by minimizing the mean square error between the filtered signal and the other signal.…”

Complex analytic signals applied on time delay estimation

“…If the condition for the perfect coherence is not satisfied, i.e., if ⎪γ s 1 s 2 (ω)⎪ ≠ 1, then the function SNR TD (ω) characterizes the estimation performance. As shown in [24], the cross-correlation function of the narrow-band signal has a number of neighboring peaks (local maximums) with approximately the same magnitude (quasi-periodic function). In case of small SNR, the reliability of cross-correlation peak determination (corresponding to the true value of differential delay) decreases.…”

“…A lot of algorithms presented to determine differential delay are based on the cross-correlation function of sensor signals. A method to determine differential delay based on FIR coefficient computation of one of the signals is presented in [24]. FIR filter coefficients are determined by minimizing the mean square error between the filtered signal and the other signal.…”

A novel online adaptive time delay identification technique

“…Since the projection strategy given in (19) ensuresτ (t) ∈ Θ s ; thus, it follows thatτ (t) ∈ L ∞ . Hence, from (12), it follows that q(·) ∈ L ∞ . Since a * (t) is a function of the bounded signals, and q(·) is a measurable bounded signal, from (16), it follows that .…”

“…Zhang and Li [11] analyzed the timevarying communication delay and proposed a time delay identification method based on steepest descent algorithm. Wen et al proposed a new symmetrical adaptive structure to solve the problem of time delay identification in noisy environment and developed a stochastic gradient algorithm to calculate the optimum solution [12]. Shaltaf presented a neuro-fuzzy technique for identification of time delay embedded within a received noisy and delayed replica of a known reference signal [13].…”

Practical continuous nonsingular terminal sliding mode control of a cable-driven manipulator developed for aerial robots

“…The core idea of TDE is to obtain the unknown dynamics at present through the state at previous moment of the system because there is only a small change in the system state as long as the time interval is fairly small (usually several sample times). TDE has been widely used in lots of research fields, such as unbiased filtering of signals, 18 parameter and state estimation 19–21 and control of nonlinear systems. 22,23 TDE in the manipulator controller can effectively reduce the dependence on the system dynamics model, and when combined with other control methods, thus, a simple and satisfactory controller is obtained.…”

Complex analytic signals applied on time delay estimation

“…If the condition for the perfect coherence is not satisfied, i.e., if ⎪γ s 1 s 2 (ω)⎪ ≠ 1, then the function SNR TD (ω) characterizes the estimation performance. As shown in [24], the cross-correlation function of the narrow-band signal has a number of neighboring peaks (local maximums) with approximately the same magnitude (quasi-periodic function). In case of small SNR, the reliability of cross-correlation peak determination (corresponding to the true value of differential delay) decreases.…”

“…A lot of algorithms presented to determine differential delay are based on the cross-correlation function of sensor signals. A method to determine differential delay based on FIR coefficient computation of one of the signals is presented in [24]. FIR filter coefficients are determined by minimizing the mean square error between the filtered signal and the other signal.…”

A novel online adaptive time delay identification technique

“…Since the projection strategy given in (19) ensuresτ (t) ∈ Θ s ; thus, it follows thatτ (t) ∈ L ∞ . Hence, from (12), it follows that q(·) ∈ L ∞ . Since a * (t) is a function of the bounded signals, and q(·) is a measurable bounded signal, from (16), it follows that .…”

“…Zhang and Li [11] analyzed the timevarying communication delay and proposed a time delay identification method based on steepest descent algorithm. Wen et al proposed a new symmetrical adaptive structure to solve the problem of time delay identification in noisy environment and developed a stochastic gradient algorithm to calculate the optimum solution [12]. Shaltaf presented a neuro-fuzzy technique for identification of time delay embedded within a received noisy and delayed replica of a known reference signal [13].…”

Practical continuous nonsingular terminal sliding mode control of a cable-driven manipulator developed for aerial robots

“…The core idea of TDE is to obtain the unknown dynamics at present through the state at previous moment of the system because there is only a small change in the system state as long as the time interval is fairly small (usually several sample times). TDE has been widely used in lots of research fields, such as unbiased filtering of signals, 18 parameter and state estimation 19–21 and control of nonlinear systems. 22,23 TDE in the manipulator controller can effectively reduce the dependence on the system dynamics model, and when combined with other control methods, thus, a simple and satisfactory controller is obtained.…”

Complex analytic signals applied on time delay estimation

“…If the condition for the perfect coherence is not satisfied, i.e., if ⎪γ s 1 s 2 (ω)⎪ ≠ 1, then the function SNR TD (ω) characterizes the estimation performance. As shown in [24], the cross-correlation function of the narrow-band signal has a number of neighboring peaks (local maximums) with approximately the same magnitude (quasi-periodic function). In case of small SNR, the reliability of cross-correlation peak determination (corresponding to the true value of differential delay) decreases.…”

“…A lot of algorithms presented to determine differential delay are based on the cross-correlation function of sensor signals. A method to determine differential delay based on FIR coefficient computation of one of the signals is presented in [24]. FIR filter coefficients are determined by minimizing the mean square error between the filtered signal and the other signal.…”

A novel online adaptive time delay identification technique

“…Since the projection strategy given in (19) ensuresτ (t) ∈ Θ s ; thus, it follows thatτ (t) ∈ L ∞ . Hence, from (12), it follows that q(·) ∈ L ∞ . Since a * (t) is a function of the bounded signals, and q(·) is a measurable bounded signal, from (16), it follows that .…”

“…Zhang and Li [11] analyzed the timevarying communication delay and proposed a time delay identification method based on steepest descent algorithm. Wen et al proposed a new symmetrical adaptive structure to solve the problem of time delay identification in noisy environment and developed a stochastic gradient algorithm to calculate the optimum solution [12]. Shaltaf presented a neuro-fuzzy technique for identification of time delay embedded within a received noisy and delayed replica of a known reference signal [13].…”

Practical continuous nonsingular terminal sliding mode control of a cable-driven manipulator developed for aerial robots

“…The core idea of TDE is to obtain the unknown dynamics at present through the state at previous moment of the system because there is only a small change in the system state as long as the time interval is fairly small (usually several sample times). TDE has been widely used in lots of research fields, such as unbiased filtering of signals, 18 parameter and state estimation 19–21 and control of nonlinear systems. 22,23 TDE in the manipulator controller can effectively reduce the dependence on the system dynamics model, and when combined with other control methods, thus, a simple and satisfactory controller is obtained.…”