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A variable stepsize gradient adaptive lattice algorithm for multiple sinusoidal interference cancelation
EURASIP Journal on Advances in Signal Processing volumeÂ 2013, ArticleÂ number:Â 106 (2013)
Abstract
In passive sonar, adaptive algorithms can be used to cancel strong sinusoidal selfinterferences. In order to correctly recover lowpower target signals during the early stages of processing, these adaptive algorithms must provide fast convergence and, at the same time, narrow notches at the frequencies of the sinusoids. In this respect, the gradient adaptive lattice (GAL) algorithm is a very attractive choice. However, the GAL algorithm with a constant stepsize parameter has to compromise between the convergence rate and notch bandwidths. Therefore, in this article, we propose a variable stepsize scheme for the GAL algorithm that can achieve both a fast convergence rate and narrow notches. Simulation results demonstrate the efficiency of the proposed algorithm compared to both the conventional GAL algorithm and transversal adaptive filter combined with the variable stepsize scheme.
1 Introduction
The performance of passive sonar is easily degraded by strong selfinterference originating from the machinery of its own ship [1]. To cancel this selfinterference composed of multiple sinusoids, adaptive noise cancelation (ANC) techniques can be used [2]. An ANC algorithm should be able to form narrow notches in order to filter out selfinterference without distorting the lowpower target signal in a passive sonar signal [3]. Among the various adaptive algorithms, the normalized least mean square (NLMS) algorithm is most widely used due to its computational simplicity and ease of implementation [4]. However, the least mean square (LMS)type transversal adaptive filters produce notch bandwidths proportional to interference amplitudes [3, 5], which can result in the distortion of the target signal in passive sonar especially when the interference has strong amplitude.
The gradient adaptive lattice (GAL) algorithm has also been widely used for ANC [6, 7]. Unlike the LMStype transversal filters, it produces notches whose bandwidths are independent of interference amplitudes [5]. Thus, the GAL algorithm is more desirable than the LMStype transversal filter in terms of the distortion in the recovered target signal. However, the GAL algorithm has the property that the stepsize parameter reflects a tradeoff between the convergence rate and the notch bandwidths, and thus, relatively wide notch bandwidths should be allowed to achieve fast convergence rate. To achieve both fast convergence rate and narrow notch bandwidth, a variable stepsize (VSS) scheme can be considered. Many VSS schemes have been developed mainly for the transversal filters employing the LMS and affine projection algorithms [8â€“14]. However, these VSS schemes are not directly applicable to the GAL algorithm due to different structure and convergence behaviors. Previously, a variable stepsize filteredx GAL (VSSFxGAL) algorithm was presented by the authors to obtain a fast adaptive algorithm for active noise control applications [15]. To estimate parameters for the stepsize control in practical situations, [15] uses the assumption that the system is in a converged state, which, however, can be problematic especially when the system is in a timevarying environment.
In this article, we revise the previous VSSFxGAL algorithm to make it suitable for the cancelation of selfinterference in passive sonar. The proposed VSS scheme takes into account the fact that the target signal needs to be preserved after canceling the strong selfinterference. Thus, the VSS scheme is developed from a condition that the target signal power is recovered at the error output. As a result, the proposed algorithm achieves both fast convergence and narrow notches at the sinusoid frequencies. Convergence analysis in the mean square sense is presented to prove the stability and steadystate performance. Simulation results corroborate the analyses in various environments.
The rest of the article is organized as follows. Section 2 starts with a presentation of the ANC configuration. Then, the classical GAL algorithm is introduced, followed by the derivation and analysis of the proposed VSSGAL algorithm. Section 3 presents the parameter estimation for real implementation. The simulation results are presented in Section 4, comparing the proposed VSSGAL algorithm from the GAL algorithm and the variable stepsize NLMS (VSSNLMS) algorithm. Finally, in Section 5, the main results of this work are discussed, and the conclusions are drawn.
2 Variable stepsize GAL algorithm
The structure of the GAL noise cancelation algorithm is shown in Figure 1. The target signal s(n) is corrupted by the uncorrelated interference v _{1}(n). The purpose of the GAL algorithm is to adaptively match the interference v _{1}(n) in the primary signal d(n)=s(n)+v _{1}(n) by linearly combining the backward prediction errors obtained from the reference signal v _{2}(n).
2.1 GAL algorithm
Consider an M thorder lattice predictor specified by the recursive equations [6]:
where f _{ m }(n) and b _{ m }(n) denote the M th stage forward and backward prediction errors at time n, respectively, and Îº _{ m }(n) is the reflection coefficient. The backward prediction errors are orthogonal to each other as
where {\mathrm{\xce\xb7}}_{m}\left(n\right)=E\left\{{b}_{m}^{2}\left(n\right)\right\} denotes the power of the m th stage backward prediction error. It can be recursively estimated using the singlepole lowpass filter as {\stackrel{\xcc\u201a}{\mathrm{\xce\xb7}}}_{m}\left(n\right)=\mathrm{\xce\xbb}{\stackrel{\xcc\u201a}{\mathrm{\xce\xb7}}}_{m}\left(n\xe2\u02c6\u20191\right)+\left(1\xe2\u02c6\u2019\mathrm{\xce\xbb}\right){b}_{m}^{2}\left(n\right), where Î» is a smoothing factor. In the classical GAL algorithm [6], the m th stage reflection coefficient is updated as
where Î¼ is the stepsize parameter and {\mathrm{\xce\xbe}}_{m}\left(n\right)=E\left\{{b}_{m}^{2}\left(n\xe2\u02c6\u20191\right)+{f}_{m}^{2}\left(n\right)\right\} is the power of both the m th stage forward and delayed backward prediction errors, which can also be recursively estimated. The update equation for the m th stage regression coefficient w _{ m }(n) is expressed as
where e _{ m }(n) is the m the stage error signal and e _{âˆ’1}(n)=d(n) is the initial error signal.
2.2 Variable stepsize GAL algorithm
To derive the variable stepsize GAL algorithm, we assume that the adaptive filter has converged to a certain degree [9, 15]. Then, due to the orthogonality of the backward prediction errors [6], we can establish the approximation:
where e _{ Mâˆ’1}(n) denotes the (Mâˆ’1)th stage error signal. Using this approximation, the update equation in Eq. (5) can be rewritten in vector notations as
where w(n)=[w _{0}(n), â€¦, w _{ Mâˆ’1}(n)]^{T} and b(n)=[b _{0}(n), â€¦, b _{ Mâˆ’1}(n)]^{T} are the regression coefficients vector and the backward prediction errors vector, respectively, and Î£(n)=d i a g{Î· _{0}(n), â€¦, Î· _{ Mâˆ’1}(n)} is a diagonal matrix. The positive scalar Î¼(n) in Eq. (8) denotes the variable stepsize parameter.
Using the regression coefficients at time n, the a posteriori error signal can be defined as
Substituting Eq. (8) into Eq. (10), we have
The target signal in passive sonar should be recovered with minimum distortion at the filter output. To this end, we can find a variable stepsize parameter Î¼(n) that satisfies the following condition:
where {\mathrm{\xcf\u0192}}_{\mathrm{\xce\mu}}^{2}\left(n\right)=E\left\{{\mathrm{\xce\mu}}^{2}\left(n\right)\right\} and {\mathrm{\xcf\u0192}}_{s}^{2}\left(n\right)=E\left\{{s}^{2}\left(n\right)\right\} are the powers of the a posteriori error signal and target signal, respectively. Thus, using the condition in Eq. (12), the target signal power is recovered at the a posteriori error signal. Now, squaring Eq. (11) and taking the expectations under the independence assumption [16], we obtain
where {\mathrm{\xcf\u0192}}_{{e}_{M\xe2\u02c6\u20191}}^{2}\left(n\right)=E\left\{{e}_{M\xe2\u02c6\u20191}^{2}\left(n\right)\right\} is the power of the (Mâˆ’1)th stage error signal. Using the orthogonality of the backward prediction errors in Eq. (3), the terms in the lefthand side of Eq. (13) can be simplified as
and
In the above simplifications, we used the approximation E\left\{{b}_{i}^{2}\left(n\right){b}_{j}^{2}\left(n\right)\right\}\xe2\u2030\u02c6E\left\{{b}_{i}^{2}\left(n\right)\right\}E\left\{{b}_{j}^{2}\left(n\right)\right\}. Hence, Eq. (13) can be rewritten as
By solving Eq. (16), we obtain the variable stepsize parameter at time n, as given by
This variable stepsize parameter replaces the stepsize parameters in Eqs. (4) and (5). In general, the power of the interference in the sonar signal is much stronger than that of the target signal. Thus, at the initial state, we have {\mathrm{\xcf\u0192}}_{{e}_{M\xe2\u02c6\u20191}}^{2}\left(n\right)\xe2\u2030\xab{\mathrm{\xcf\u0192}}_{s}^{2}\left(n\right), and the stepsize parameter will be determined as Î¼ _{ v s s }(n)â‰ˆ1/M, which leads to fast convergence. When the algorithm approaches the steadystate, we can expect that {\mathrm{\xcf\u0192}}_{{e}_{M\xe2\u02c6\u20191}}^{2}\left(n\right)\xe2\u2030\u02c6{\mathrm{\xcf\u0192}}_{s}^{2}\left(n\right), and as a result, a fairly small stepsize parameter will be used. Thus, narrow notch bandwidths will be attained.
2.3 Convergence analysis
The mean square behavior of the VSSGAL algorithm can be analyzed by evaluating secondorder moments of the regression coefficient errors. Suppose that the primary signal is modeled as
where {\mathbf{w}}^{o}={\left[{w}_{0}^{o},\phantom{\rule{0.3em}{0ex}}\xe2\u20ac\xa6,\phantom{\rule{0.3em}{0ex}}{w}_{M\xe2\u02c6\u20191}^{o}\right]}^{T} is the impulse response of the unknown system with length M. The update equation in Eq. (5) can thus be written in terms of the m th stage regression coefficient error, {\stackrel{~}{w}}_{m}\left(n\right)={w}_{m}^{o}\xe2\u02c6\u2019{w}_{m}\left(n\right), as
Then, we have
Using the orthogonality of the backward prediction errors in Eq. (3) and the independence condition [16], the second and third terms in the righthand side of the above equation can be approximated, respectively, as
and
where {\mathrm{\xcf\u0192}}_{{e}_{m}}^{2}\left(n\right)=E\left\{{e}_{m}^{2}\left(n\right)\right\} is the power of the m th stage error signal. In the above equation, the same approximation is used in Eq. (15). Using Eqs. (21) and (22), Eq. (20) can be rewritten as
Thus, we readily see that E\left\{{\stackrel{~}{w}}_{m}^{2}\left(n\right)\right\}\xe2\u02c6\u2019E\left\{{\stackrel{~}{w}}_{m}^{2}(n\xe2\u02c6\u20191)\right\}<0 is achieved if the variable stepsize parameter Î¼(n) is bounded as
Since the filter order M should be twice as many as the number of sinusoids for sinusoidal interference cancelation, the variable stepsize parameter in Eq. (17) always lies within the stability bound.
Next, we will show that the VSSGAL algorithm recovers the target signal power at the filter output. Substituting Eq. (18) into Eq. (9), we have
where \stackrel{~}{\mathrm{w}}\left(n\right)={\left[{\stackrel{~}{w}}_{0}\left(n\right),\phantom{\rule{0.3em}{0ex}}\xe2\u20ac\xa6,\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{w}}_{M\xe2\u02c6\u20191}\left(n\right)\right]}^{T}. Squaring Eq. (25) and taking expectations, we obtain
Again, using the orthogonality of the backward prediction errors and the independence assumption [16], Eq. (26) can be rewritten as
where \mathbf{K}\left(n\xe2\u02c6\u20191\right)=E\left\{\stackrel{~}{\mathrm{w}}\left(n\xe2\u02c6\u20191\right){\stackrel{~}{\mathrm{w}}}^{T}\left(n\xe2\u02c6\u20191\right)\right\}. If the stepsize parameter is properly bounded, as in Eq. (24), we can achieve E\left\{{\stackrel{~}{w}}_{m}^{2}\left(\mathrm{\xe2\u02c6\u017e}\right)\right\}=0. Then, we also have t r{Î£(âˆž)K(âˆž)}=0. Thus, the variable stepsize parameter in Eq. (17) guarantees that
Thus, it has been proven that the VSSGAL algorithm in steadystate recovers the target signal power at the final stage of the lattice filter.
3 Parameter estimation
To determine the variable stepsize parameter Î¼ _{ v s s }(n), we need a target signal power {\mathrm{\xcf\u0192}}_{s}^{2}\left(n\right), which is not available in realworld applications. To solve this problem, we approximate Eq. (26) as
The crosscorrelation between the backward prediction errors vector and the (Mâˆ’1)th stage error signal is given by
Now, the target signal power {\mathrm{\xcf\u0192}}_{s}^{2}\left(n\right) can be estimated by substituting Eq. (30) into Eq. (29):
In practice, Î¾ _{ m }(n), Î· _{ m }(n), {\mathrm{\xcf\u0192}}_{{e}_{M\xe2\u02c6\u20191}}^{2}\left(n\right), and r(n) can be estimated using a singlepole lowpass filter. Without a loss of generality, a variable smoothing factor Î» _{1}(n)=1âˆ’Î¼ _{ v s s }(n) can be used. However, Î¼ _{ v s s }(n) is not available until these parameters are obtained. Thus, we use the previous variable stepsize parameter Î¼ _{ v s s }(nâˆ’1) to estimate parameters Î¾ _{ m }(n) and Î· _{ m }(n), with Î¼ _{ v s s }(âˆ’1)=1/M. On the other hand, {\mathrm{\xcf\u0192}}_{{e}_{M\xe2\u02c6\u20191}}^{2}\left(n\right) and r(n) are estimated using a fixed smoothing factor 0<Î» _{2}<1. The following equations denote the estimators.
The estimate {\stackrel{\xcc\u201a}{\mathrm{\xcf\u0192}}}_{{e}_{M\xe2\u02c6\u20191}}^{2}\left(n\right) can be smaller than {\stackrel{\xcc\u201a}{\mathrm{\xcf\u0192}}}_{s}^{2}\left(n\right), which can result in negative Î¼ _{ v s s }(n). To avoid this, we use the absolute value of Eq. (17). Hence, the variable stepsize parameter is determined as
4 Simulation results
Computer simulations were conducted to evaluate the proposed VSSGAL algorithm. In sonar signal processing, both the target signal s(n) and interference source v(n) consist of multiple sinusoids and ambient noise Ï•(n). To simulate this, we express the target signal and interference source as
where I is the number of the sinusoids and A _{ i }, f _{ i }, and Î¸ _{ i } are the amplitude, frequency, and phase of the i th sinusoid, respectively. We assumed that the interference v _{1}(n) and reference signal v _{2}(n) had sinusoidal components with identical frequencies but different amplitudes and phases. In all simulations, the sampling rate was f _{ s }=8 kHz, the interference source comprised ten sinusoids of frequencies 250, 630, 1,020, 1,380, 1,890, 2,100, 2,530, 2,950, 3,460, and 3,700 Hz. The target signal comprised ten sinusoids of frequencies 280, 680, 1,080, 1,500, 1,800, 2,150, 2,830, 3,100, 3,500, and 3,730 Hz and the ambient noise which was an AR(2) process with the transfer function 1/(1âˆ’0.9z ^{âˆ’1}+0.3z ^{âˆ’2}). Figure 2 shows power spectral densities of the reference and primary signals where the interferencetosignal ratio (ISR) is 20 dB.
Performance was measured using the mean square error (MSE) defined as \mathit{\text{MSE}}\left(n\right)=E\left\{{e}_{M\xe2\u02c6\u20191}^{2}\left(n\right)\right\} and the excess mean square error (EMSE) defined as E M S E(n)=E{(e _{ Mâˆ’1}(n)âˆ’s(n))^{2}} [17]. We measured the EMSE to show how closely the algorithm recovers the target signal at the error output. All the MSE and EMSE were obtained by averaging 100 independent trials.
In Figure 3, the EMSE of the proposed VSSGAL algorithm is compared with those of the conventional GAL algorithm obtained using two different stepsize parameters, Î¼=0.015 and Î¼=0.0028. Reference and primary signals shown in Figure 2 were used, and the filter order was M=20. To assess tracking ability, the phases of the interference sinusoids were randomly changed at the 5,000th sample. The results in the figure show that the convergence rate of the proposed algorithm is as fast as that of the conventional GAL algorithm with Î¼=0.015 and the steadystate EMSE is as low as the one obtained using Î¼=0.0028. We also show the averaged stepsize parameter of the VSSGAL algorithm in Figure 3, which clearly indicates that the proposed VSS scheme desirably adjusts the stepsize parameter according to the state of the filter response.
In Figures 4, 5, 6, and 7, we compared the performance of the proposed VSSGAL algorithm with those of the VSSNLMS algorithm in [9] and the VSSGAL algorithm in [15]. Simulation environments were the same as Figure 3. Figure 4 shows the MSE curves. It is shown that the proposed VSSGAL algorithm converges faster than the VSSNLMS algorithm, and the steadystate MSE of the proposed VSSGAL algorithm is slightly higher than that of the VSSNLMS algorithm. The previous VSSGAL algorithm in [15], on the other hand, shows slow convergence especially when the interference was changed at 5,000 samples. The reason is that, since it requires an assumption that the adaptive filter has converged well enough to estimate the target signal power, the estimated parameters during the transient state are biased ones. The EMSE curves in Figure 5 more clearly demonstrate the superiority of the proposed VSSGAL algorithm over the other algorithms. Figure 5 shows that the proposed VSSGAL algorithm produces significantly lower steadystate EMSE than both the VSSNLMS and the previous VSSGAL algorithms, which indicates that the proposed VSSGAL algorithm is able to recover the target signal with much lower distortion than the other algorithms. This is because the proposed VSSGAL algorithm produces accurate notches with narrow bandwidths which are independent of the amplitudes of sinusoidal interferences and proportionate only to the stepsize parameter.
In [5], it was shown that the i th notch bandwidth of the NLMS transversal filter for multiple sinusoidal interference can be approximated as
where \stackrel{~}{\mathrm{\xce\xbc}}=\mathrm{\xce\xbc}/{\mathrm{\xcf\u0192}}_{x}^{2} is the normalized stepsize parameter and {\mathrm{\xcf\u0192}}_{x}^{2} is the power of the reference signal. Thus, the notch bandwidth of the NLMS transversal filter is proportional to the amplitude of the sinusoidal interference A _{ i } and the filter order M. On the other hand, the i th notch bandwidth of the GAL filter can be approximated as [5]
where the notch bandwidth is independent of the amplitude of the sinusoidal interference and it is proportional only to the stepsize parameter Î¼. In the VSS scheme, a small stepsize parameter is generally used at the steadystate so that narrow notches can be provided.
To compare the notch behaviors, we evaluated the transfer function from d(n) to e(n) at the steady state. The results are presented in Figure 6. Also, the power spectral densities of the recovered target signals are presented in Figure 7. The transfer functions in Figure 6 show that the VSSNLMS algorithm distorts the frequency response around the frequencies of sinusoidal interferences with large amplitudes, which is clearly visible at 250 and 1,020 Hz. As a result, the VSSNLMS algorithm produced high EMSE. On the other hand, both the previous and the proposed VSSGAL algorithms produced insignificant distortions at frequencies other than the interference frequencies. However, the notches produced by the previous VSSGAL algorithm were not sufficiently accurate so that the interferences were not completely removed, which can be seen from the steadystate power spectral density in Figure 7(b). Consequently, the proposed VSSGAL algorithm achieved the lowest EMSE among the algorithms, and the target signal was recovered with the smallest distortion.
Finally, to confirm the performance in various environments, we measured the steadystate MSE and EMSE according to the ISR and filter order. The results are summarized in Tables 1 and 2. EMSE results indicate that the proposed VSSGAL algorithm recovers the target signal power with much less distortions than both the VSSNLMS algorithm and the VSSGAL algorithm in [15].
5 Conclusions
We proposed a variable stepsize scheme for the GAL algorithm for cancelation of sinusoidal interference. The proposed VSS scheme was designed to recover the target signal from within the error signal. Simulation results showed that the proposed algorithm achieved a fast convergence rate, good tracking ability, and low steadystate EMSE. Compared to the VSSNLMS algorithm, it formed narrow notches at the interference frequencies, so it could recover the target spectrum with significantly smaller distortions than the VSSNLMS algorithm.
Abbreviations
 ANC:

Adaptive noise cancelation
 EMSE:

Excess mean square error
 GAL:

Gradient adaptive lattice
 ISR:

Interferencetosignal ratio
 MSE:

Mean square error
 NLMS:

Normalized least mean square
 VSS:

Variable step size
 VSSGAL:

Variable stepsize gradient adaptive lattice
 VSSNLMS:

Variable stepsize normalized least mean square.
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Kim, Sw., Park, Yc. & Youn, D.H. A variable stepsize gradient adaptive lattice algorithm for multiple sinusoidal interference cancelation. EURASIP J. Adv. Signal Process. 2013, 106 (2013). https://doi.org/10.1186/168761802013106
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DOI: https://doi.org/10.1186/168761802013106