FAST ADAPTIVE DETECTION OF SINUSOIDAL SIGNALS USING VARIABLE DIGITAL FILTERS AND ALL-PASS FILTERS

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1 FAST ADAPTIVE DETECTION OF SINUSOIDAL SIGNALS USING VARIABLE DIGITAL FILTERS AND ALL-PASS FILTERS Keitaro HASHIMOTO and Masayuki KAWAMATA Department of Electronic Engineering, Graduate School of Engineering Tohoku University Aoba-yama 05, Sendai, , Japan Phone: , Fax: ABSTRACT This paper proposes an adaptive line enhancer (ALE) using a variable band-pass (BP) filter and an all-pass filter. The proposed ALE has the high convergence speed and the small steady-state value of the squared error. The all-pass filter outputs the signal which consists of a sinusoidal signal which has the same phase as that of the input signal to the ALE and a white noise which has no correlation with that of the input signal to the ALE. The variable BP filter has two parameters which control its center frequency and bandwidth. In the proposed ALE, both of the center frequency and the bandwidth are adaptively adjusted using least mean squares algorithm. As the reference signal, the input signal to the ALE is adopted to adjust the center frequency and the output signal of the all-pass filter is adopted to adjust the bandwidth. Numerical examples show that the proposed ALE has the high convergence speed and the small steady-state value of the squared error. 1. INTRODUCTION In many signal processing applications, such as communications and radar, it is necessary to detect and enhance sinusoidal signals corrupted by broadband noise. Such signals are detected and enhanced using an adaptive line enhancer (ALE). This paper proposes a new ALE using a variable digital filter and an all-pass filter. The ALE using a variable band-pass (BP) filter has been proposed by Hagiwara and Kawamata [1]. The variable BP filter used in Ref. [1] has two parameters which control its center frequency and bandwidth. The ALE proposed in Ref. [1] adaptively adjusts only the center frequency of a variable BP filter using normalized recursive least mean squares (NRLMS) algorithm. The convergence speed of the ALE using a wideband variable BP filter is higher than that of the ALE using a narrow-band variable BP filter. However the steady-state value of the squared error of the ALE using a wide-band variable BP filter is larger than that of the ALE using a narrow-band variable BP filter. Therefore it is desirable to adaptively adjust the bandwidth of a variable BP filter. This paper proposes the ALE using a variable BP filter which adaptively adjusts both of the center frequency and the bandwidth.. VARIABLE DIGITAL FILTERS Variable digital filters are frequency selective filters which can be tuned their frequency characteristics, such as the cutoff (center) frequency and the bandwidth by adjusting some parameters []. They are usually designed by employing the frequency transformation of Constantinides [3] based on the substitution where T (z) is a first- or second-order all-pass transfer function. It is possible to obtain a variable BP filter by applying Eq. (1) to the transfer function H LP (z) of a low-pass (LP) filter with a cutoff frequency ω cp. For LP to BP transformation, where ω 0 is the center frequency, and ω c1 and ω c are cutoff frequencies. The center frequency and the bandwidth of a variable BP filter are tuned by adjusting the parameters α and β. The variable BP filters have always zero phase shift at the center frequency ω 0. This property is used in the next section. We consider second-order variable BP filters. The transfer function of a prototype LP filter is given by Applying Eq. () to Eq. (5), the transfer function of a secondorder variable BP filter is obtained as follows: 73

2 3. ADAPTIVE LINEENHANCER USING VARIABLE DIGITAL FILTERS AND ALL-PASS FILTERS In this section, we discuss the structure of the ALE using a variable BP filter and an all-pass filter and the error surface for the ALE. We also discuss adaptive algorithm for the adaptation of the parameters α and β which control the center frequency and the bandwidth of the variable BP filter, respectively Structure of the Proposed Adaptive Line Enhancer Figure 1 shows the structure of the proposed ALE using a variable BP filter and an all-pass filter. The input signal x k is σ n. where n k is a white noise of zero mean and variance The transfer function of a second-order all-pass filter H AP ( z, ξ ) is given by where the parameter ξ controls the phase response of the all-pass filter. The all-pass filter outputs the signal which consists of a sinusoidal signal which has the same phase as that of the input signal x k and a white noise which has no correlation with that of the input signal x k. The delay, denoted by z M in Figure 1, removes the correlation between the white noise of the input signal x k and that of the input signal to the all-pass filter, while a phase shift is introduced between the sinusoidal components of these two signals [4]. The all-pass filter adaptively corrects the phase shift. Thus the parameter ξ is adaptively adjusted to correct the phase shift of a sinusoidal signal using NRLMS algorithm. As a result, y ξ, k, the output signal of the all-pass filter consists of a sinusoidal signal which has the same phase as that of the input signal x k and the white noise n k which has no correlation with n k of the input signal x k. The variable BP filter detects and enhances a sinusoidal Figure 1: Structure of ALE using a variable BP filter and an all-pass filter signal. The parameters α and β of the variable BP filter are adaptively adjusted. α is adaptively adjusted as proposed in Ref. [1], where NRLMS algorithm is used. Then x k is adopted as the reference signal. When β is adaptively adjusted using NRLMS algorithm, it is necessary to remove the correlation between the white noise of the reference signal and that of the input signal to the variable BP filter. If the input signal to the variable BP filter is delayed to remove the correlation, the variable BP filter can not detect and enhance a sinusoidal signal. The reason is that the variable BP filter can not correct the phase shift of a sinusoidal signal because it has always zero phase shift at the center frequency ω 0. In the proposed ALE, y ξ, k, the output signal of the all-pass filter is adopted as the reference signal to adaptively adjust β. Thus the parameters α and β converges to the optimal value to detect and enhance a sinusoidal signal. The enhanced sinusoidal signal is available at the output y k of the variable BP filter. 3.. The Convergence Speed and the Steady-State Value of the Squared Error of the Proposed ALE We discuss the convergence speed and the steady-state value of the squared error of the proposed ALE using error surfaces. The mean-square error (MSE) for α is where S x (z) is the input power spectrum, consisting of the power spectrum of s x and n x in Eq. (11) [5,6]. Substituting Eq. (6) into Eq. (14), the MSE is obtained 74

3 (a) α = α = (b) α = 0, Figure 3: Error surface for β Figure : Error surface for α Figure 4: Error surface for ξ MSE for β is where H ( e jω s, α, β ) is the frequency response of the variable BP filter. Solving the equation E [ eα,k ]/ α =0 with respect to α, the value of α that minimizes Eq. (15) is Comparing this result with Eq. (3), the optimal value of the center frequency ω 0 becomes ω 0 = ωs. Figure shows the error surfaces, that is, E[ e α,k ] in Eq. (14) versus α for different β of a second-order variable BP filter. A, ω s and σ n are 1.0, 0.4π and 0.1 respectively. As illustrated in Figure, the error surface for α is unimodal. Therefore a gradient-based search technique is applicable for adjusting the parameter α. We observe from Figure that the error surface becomes flat as β becomes large in areas away from the minimum, while the minimum value of the error surface becomes small. This suggests that there is a relation between the convergence speed of α and the steady-state value of the squared error for β as follows: When β is small, the convergence speed of α is high, but the steady-state value of the squared error of the ALE is large. When β is large, the steady-state value of the squared error of the ALE is small, but the convergence speed of α is low. Hence in the proposed ALE, β is also adaptively adjusted to achieve the high convergence speed of α and the small steady-state value of the squared error. The Substituting Eq. (6) and Eq. (13) into Eq. (17), the MSE is obtained jω s where H AP( e, ξ ) is the frequency response of the all-pass filter and p and q are the poles of H ( z, α, β ). Figure 3 shows the error surfaces, that is, E[ e β,k ] in Eq. (18) versus β for different α and for M = 47. As illustrated in Figure 3, the error surface for β is unimodal for α = α or α α, but it is not unimodal for α which is far from α. We can adopt a gradient-based search technique to adjust β if we know whether α is far from α or not. Thus a manipulation to set β at the small value when α is far from α is need to adopt a gradient-based search technique. Therefore β becomes large for α = α, and β becomes small for α α when β is adaptively adjusted using a gradient-based search technique. From this, the proposed ALE has the high convergence speed and the small steady-state value of the squared error. We will next consider the error surface for the all-pass filter in the proposed ALE. The MSE for ξ is given by 75

4 Substituting Eq. (13) into Eq. (19), the MSE is obtained Figure 4 shows the error surface, that is, E[ e ξ,k ] in Eq. (3.) versus ξ of the all-pass filter H AP ( z, ξ ) for M = 47. As illustrated in Figure 4, the error surface is not unimodal. If the point of ξ = 1. 0 is connected with that of ξ = 1. 0, the error surface for ξ is unimodal. Thus a gradient-based search technique is also applicable for adjusting ξ Adaptive Algorithm We adopt the normalized recursive least mean squares (NRLMS) algorithm [7]. The reason is that NRLMS algorithm needs few calculations. First the all-pass filter is adaptively adjusted. Applying NRLMS algorithm, ξ is updated as follows: where µ ξ is a step size. Then the proposed method needs a manipulation given by Second the variable BP filter is adaptively adjusted. Applying NRLMS algorithm, α and β are updated as follows: Figure 5: Transient behavior of ξ As we discuss in Section 3., the error surface for β is not unimodal for α which is far from the optimal value. The input autocorrelation R α is large when α is the optimal value and small when α is far from the optimal value. From this, we judge that α is far from the optimal value by examining whether R α is smaller than a threshold value or not. However, it is difficult to decide a threshold value because the value of R α is too large when α is the optimal value and depends on the magnitude of a sinusoidal signal. It is easier to decide a threshold value to judge using 1 / Rα than using R α. Therefore we adopt 1 / Rα to judge whether α is far from the optimal value or not. The manipulation to judge whether α is far from the optimal value or not is where R, th 1/ α is a threshold value. 4. COMPUTER SIMULATIONS We have realized an ALE using a second-order variable BP filter and a second-order all-pass filter. The transfer function of a prototype LP filter with cutoff frequency ω cp = 0.1 π is where R α and R β are input autocorrelation functions, γ α and γ β are forgetting factors, and µ α and µ β are step sizes. From Eq. (6), Eq. (8) can be replaced by The frequency ω s of the sinusoidal signal is 0. π, which is switched to 0.7 π after,500 sample points. We show the values of the parameters used in the simulation in Table 1. Figure 5 shows the transient behavior of the parameter ξ. Figure 6 shows the transient behavior of α and the center frequency. We observe from Figure 6 that the convergence speed of α in the proposed ALE is higher than that of α in Ref. [1]. Figure 7 shows the transient behavior of 1 / Rα. We observe from Figure 7 that 1 / Rα is larger than 1/ R α, th when α is far from the optimal value. 76

5 (a) transient behavior of α (a) transient behavior of β (b) transient behavior of the center frequency Figure 6: Transient behavior of α and the center frequency (b) transient behavior of the bandwidth Figure 8: Transient behavior of β and the bandwidth , December [] G. Stoyanov and M. Kawamata, Variable digital filters, Journal of Signal Processing, vol. 1, no. 4, pp , July Figure 7: Transient behavior of 1 / Rα Figure 8 shows the transient behavior of β and the bandwidth. We observe from Figure 8 that β becomes large when α is the optimal value, and becomes small when α is not the optimal value. The maximum value of β is restricted to 5.0 to stabilize β. The steady-state value of the squared error of the proposed ALE is and that of Ref. [1] is From this, we conclude that the proposed ALE has as the small steady-state value of the squared error as Ref. [1]. 5. CONCLUSION We have presented an ALE using a variable BP filter and an all-pass filter. In the proposed ALE, both the center frequency and the bandwidth are adaptively adjusted. Consequently, the proposed ALE achieves high convergence speed keeping the steady-state value of the squared error small. 6. REFERENCES [1] Y. Hagiwara and M. Kawamata, Adaptive detection and enhancement of sinusoidal signals using a variable IIR band-pass digital filter, Proceedings of IEEE International Symposium on Intelligent Signal Processing and Communication Systems, pp. [3] A. G. Constantinides, Spectral transformations for digital filters, Proceedings of the IEE, vol. 117, pp , August [4] S. Haykin, Adaptive Filter Theory, Prentice Hall, third edition, [5] S. D. Stearns, R. A. David and D. M. Etter, A survey of IIR adaptive filtering algorithms, Proceedings of IEEE International Symposium on Circuits and Systems, pp , May, 198. [6] C. R. Johnson Jr. and M. G. Larimore, Comments on and additions to An Adaptive Recursive LMS Filter, Proceedings of the IEEE, vol. 65, no. 9, pp , September [7] R. A. David and S. D. Stearns, Adaptive IIR algorithms based on gradient search, Proceedings of 4th Midwest Symposium on Circuits and Systems, June

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