A gradient-adaptive lattice-based complex adaptive notch filter

Transcription

1 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 DOI /s EURASIP Journal on Advances in Signal Processing RESEARCH Open Access A gradient-adaptive lattice-based complex adaptive notch filter Rui Zhu, Feiran Yang Jun Yang * Abstract This paper presents a new complex adaptive notch filter to estimate track the frequency of a complex sinusoidal signal. The gradient-adaptive lattice structure instead of the traditional gradient one is adopted to accelerate the convergence rate. It is proved that the proposed algorithm results in unbiased estimations by using the ordinary differential equation approach. The closed-form expressions for the steady-state mean square error the upper bound of step size are also derived. Simulations are conducted to validate the theoretical analysis demonstrate that the proposed method generates considerably better convergence rates tracking properties than existing methods, particularly in low signal-to-noise ratio environments. Keywords: Frequency tracking, Adaptive notch filter, Gradient-adaptive lattice, Steady-state mean square error 1 Introduction The adaptive notch filter (ANF) is an efficient frequency estimation tracking technique that is utilised in a wide variety of applications, such as communication systems, biomedical engineering radar systems [1 1]. The complex ANF (CANF) has recently gained much attention [13 0]. A direct-form poles zeros constrained CANF was first developed in [13] with a modified Gauss- Newton algorithm. A recursive least square (RLS)-based Steiglitz-McBride (RLS-SM) algorithm was also established to accelerate the convergence rate [14]. However, both algorithms are computationally complicated can result in biased estimations. To address this problem, numerous efficient unbiased least mean square (LMS)-based algorithms have been developed, such as the complex plain gradient (CPG) [15], modified CPG (MCPG) [16], lattice-form CANF (LCANF) [17], arctangent-based algorithms [18]. However, all these LMS-based algorithms generate a lower convergence rate than the RLS-based algorithms do. Moreover, the upper bound of the step size in LMSbased methods must be maintained within a limited range to ensure stability; this range depends on the eigenvalue of the correlation matrix of the input signal. These *Correspondence: jyang@mail.ioa.ac.cn The State Key Laboratory of Acoustics the Key Laboratory of Noise Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, 1, Beisihuanxilu Road, Beijing, China drawbacks limit the practical applications of LMS-based algorithms. Several normalized LMS (NLMS)-based CANF algorithms were established, including the normalized CPG (NCPG) algorithm [19] the improved simplified lattice complex algorithm [0]. However, the former may be unstable in low signal-to-noise ratio (SNR) conditions, the latter can only be used to estimate positive instantaneous frequency. In this paper, we develop a new CANF system based on the lattice algorithm [1]. Instead of the traditional gradient estimation filter, we proposed a normalized lattice predictor that makes both forward backward predictions. This scheme reduces computational complexity enhances the robustness to noise influence. Furthermore, convergence rate is improved significantly when compared with conventional gradient-based or nongradientbased methods without sacrificing tracking property. A classic ordinary differential equation (ODE) method is applied to confirm the unbiasedness of the proposed algorithm. In addition, theoretical analyses are conducted on the stable range of the step size the steady-state mean square error (MSE) under different conditions. Computer simulations are conducted to confirm the validity of the theoretical analysis results the effectiveness of the proposed algorithm. The following notations are adopted throughout this paper. j denotes square root of minus one. ln[ ] denotes 016The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, reproduction in any medium, provided you give appropriate credit to the original author(s) the source, provide a link to the Creative Commons license, indicate if changes were made.

2 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page of 11 the principal branch of the complex natural logarithm function Im{ } means taking the imaginary part of a complex value. Z{ } E{ } denote the z-transform operator statistical expectation operator, respectively. δ( ) represents the Dirac function. Asterisk denotes a complex conjugate is the convolution operator. Filter structure adaptive algorithm We consider the following noisy complex sinusoidal input signal x(n) with amplitude A, frequency ω 0 initial phase φ 0 : x(n) Ae j(ω 0n+φ 0 ) + v(n), (1) φ 0 is uniformly distributed over [ 0, π) v(n) v r (n)+jv i (n) is assumed to be a zero-mean white complex Gaussian noise process. It is assumed v r (n) v i (n) are uncorrelated zero-mean real white noise processes with identical variances. The first-order, pole-zero-constrained CANF with the following transfer function is widely used to estimate frequency ω 0 : H(z) 1 ejθ z 1 θ 1 αe jθ z is the notch frequency α represents the 1 pole-zero constrained factor determines the notch filter s 3-dB attenuation bwidth. The pole can remain in the unit circle by restricting the value of α. Wenowproposeanewstructuretoimplementthe complex notch filter. As shown in Fig. 1, the input signal x(n) is first processed by an all-pole prefilter H p (z) 1/D(z) 1/(1 + a 0 z 1 ) to obtain s 0 (n), a 0 is the coefficient of the all-pole filter. Then, a lattice predictor is employed to identify the forward backward prediction errors s 1 (n) r 1 (n), respectively.thetransform functions from s 1 (n) r 1 (n) to s 0 (n) are given by H f (z) N(z) 1 + k 0 z 1 H b (z) z 1 N (z) k 0 + z 1 (k 0 being the reflection coefficient of the lattice filter). To acquire the desired pole-zero constrained notch filter, the following relations must be satisfied: k 0 e jθ, () a 0 αk 0. (3) Thus, θ can be computed as θ Im{ln[ k 0 ] }. At this point, a normalized stochastic gradient algorithm is derived to update the reflection coefficient k 0.We consider the following cost function: J fb 1 E [ s 1 (n) + r 1 (n) ], (4) We replace cost function J fb with its instantaneous estimation, i.e., Ĵ fb 1 ( s1 (n) + r 1 (n) ). (5) By taking the derivative of Ĵ fb with respect to θ(n), we obtain Ĵ fb dĵ fb (n) dk 0 (n) dk 0 (n) dθ(n) Im{s 1(n)s 0 (n)}. (6) Considering that θ(n) is real, the adaptation equation can be written as θ(n + 1) θ(n) + μ Im{s 1 (n)s 0 (n)}/ξ(n), (7) μ is the step size the normalized signal ξ(n) can be recursively calculated as ξ(n) ρξ(n 1) + (1 ρ)s 0 (n)s 0 (n), (8) ρ denotes the smoothing factor. Table 1 shows the computational complexities of the proposed algorithm of four conventional methods [14, 16, 17, 19]. Note that the complexity of the proposed algorithm is comparable to that of LMS-based methods lower than that of NLMS-based RLS-based algorithms. 3 Convergence analysis We now use the ODE approach to analyse the convergence properties of the adaptive algorithm, which has been applied to analyse several other ANF algorithms [17, ]. Assuming that the adaptation is sufficiently slow the input signal is stationary, the associated ODEs for the proposed adaptive algorithm can be expressed as Fig. 1 Structure of a first-order complex notch filter

3 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page 3 of 11 Table 1 Complexities of the proposed algorithm of four conventional algorithms + sin&cos atan Proposed MCPG [16] LCANF[17] NCPG [19] RLS-SM [14] d dτ θ(τ) ξ 1 (τ)f (θ(τ)), (9) d ξ(τ) G(θ(τ)) ξ(τ), (10) dτ G(θ(τ)) E{s 0 (n)s 0 (n)} f (θ(τ)) Im{E[ s 0 (n)s 1 (n)] } { 1 π S x (ω)n(e jω } ) Im π π D (e jω )D(e jω ) dω { 1 π σ v N(e jω } ) Im π π D (e jω )D(e jω ) dω { } A N(e jω 0) +Im D(e jω 0) { σv } Im + A sin(ω 0 θ) 1 + α D(e jω 0) A sin(ω 0 θ) D(e jω 0 ). (11) Here, S x (ω) is the power spectral density (PSD) of x(n): S x (ω) πa δ(ω ω 0 ) + σ v [17] the transfer functions N(e jω ) 1/D(e jω ) are defined in the previous section e jω is substituted by z.sinceeq.9isthe associated ordinary differential equation of the proposed adaptive algorithm, according to [3], θ(n) will always converge to the stationary point of Eq. 9 without exception, this stationary point must satisfy d dτ θ(τ) 0. ξ(τ) is always positive; therefore, the stationary point of θ(n) converges to a solution of equation f (θ(τ)) 0. Based on Eq. 11, θ ω 0 is the sole stationary point over one period of the function. To confirm that the stationary point is stable, we choose a Lyapunov function L(τ) [ ω 0 θ(τ)]. L(τ) 0 for all τ. Meanwhile, dl(τ) dτ dl dθ dθ dτ A sin(ω 0 θ(τ))[ ω 0 θ(τ)] D(e jω 0) ξ(τ) < 0 (1) is maintained for all θ(τ) ω 0. This equation implies that L(τ) is a decreasing function of τ for ω 0 θ(τ) < π. Thus, it is proved that θ(n) canalwaysconvergetothe expected frequency ω 0 [3]. Now, we would like to compute the upper bound of step size μ. Taking the expectation on both sides of Eq. 7, we obtain { [ s0 θ(n + 1) θ(n) (n)s 1 (n) μim E ξ(n) θ(n) E{θ(n)}. Exping Eq. 8 yields ξ(n) (1 ρ) n m0 ]}, (13) ρ m s 0 (n m)s 0 (n m). (14) Taking ensemble expectations on both sides assuming that s 0 (n) is wide-sense stationary, we have [ ] n lim E[ ξ(n)] lim (1 ρ) ρ m r S0 (0) n n r S0 (0) 1 π π π m0 r S0 (0), (15) a 0 e jω S x (ω)dω A 1 + a 0 e jω 0 + σ v 1 α. (16) In each step, we consider that [4] ξ(n) r S0 (0) + ξ(n), (17) ξ(n) is the zero-mean stochastic error sequence that is independent of the input signal. By applying Eq. 17 disregarding the second-order error, we obtain 1 ξ(n) r 1 S 0 (0) r S 0 (0) ξ(n). (18) By substituting Eqs. 11, 16, 18 into Eq. 13, we get { [ ( θ(n + 1) θ(n) μim E s 0 (n)s 1 (n) r 1 S 0 (0) )]} r S 0 (0) ξ(n) { } μim E[ s 0 (n)s 1 (n)r 1 S 0 (0)] Considering the approximations μa sin(ω 0 θ(n)) 1 αe j( θ(n) ω 0 ) A + σ. 1 αe j( θ(n) ω 0 ) v 1 α sin( θ ω 0 ) θ ω 0 1 αe j( θ ω 0 ) (1 α) sin( θ ω 0 )/( θ ω 0 ) 1 ( for a small θ ω 0 ) [17], we have

4 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page 4 of 11 ( ω 0 θ(n+1) 1 μa A + 1 α 1+α σ v ) [ ω 0 θ(n)]. (19) To satisfy ω 0 θ(n + 1) < ω 0 θ(n),thestep-size μ should satisfy: 0 <μ<(1 + 1 α σ v ). (0) 1 + α A Furthermore, when SNR or α 1, we have μ (0, ], which is independent of the input. 4 Steady-state MSE analysis In this section, a PSD-based method [19, 5] is exploited to derive the accurate expressions for the steady-state MSE of the estimated frequency. As discussed in the previous section, the estimated frequency can converge to an unbiased value, i.e., lim θ(n) ω 0. Defining that θ(n) θ(n) ω 0, we obtain the following n two approximations: lim sin( θ(n)) θ(n) n lim cos( θ(n)) 1. Then, the steady-state transfer n function from s 1 (n) s 0 (n) to x(n) can be written as: H s1 (e jω 0 ) 1 ej θ(n) j θ(n) 1 αej θ(n) 1 α, (1) H s0 (e jω 0 1 ) 1 αe j θ(n) 1 1 α. () The input signal x(n) in Eq. 1 is assumed to be composed of a single frequency part Gaussian white noise. Thus, the steady-state outputs s 1 (n) s 0 (n) can be expressed as: s 1 (n) s s1 (n) + n s1 (n), (3) s 0 (n) s s0 (n) + n s0 (n), (4) n s1 (n) n s0 (n) arethecomplexgaussianpartsof s 1 (n) s 0 (n), respectively. By using Eqs. 1, we obtain s s1 (n) A θ(n)ej(ω 0n+φ 0 π/), (5) 1 α s s0 (n) Aej(ω 0n+φ 0 ). (6) 1 α By substituting Eqs. 3 4 into Eq. 7, the adaptive update equation can be rewritten as θ(n + 1) θ(n) + μ 4 u i (n), (7) i1 μ μ ξ(n) μ A /(1 α) + σ v /(1 α ), (8) u 1 (n) Im{s s0 (n)n s1 (n)}, (9) u (n) Im{n s0 (n)n s1 (n)}, (30) u 3 (n) Im{s s0 (n)s s1 (n)}, (31) u 4 (n) Im{n s0 (n)s s1 (n)}. (3) Substituting Eqs. 5 6 into Eq. 31 yields u 3 (n) A θ(n) (1 α). (33) Meanwhile, Eq. 3 can be rearranged as u 4 (n) Im(n s0 (n) ja θ(n)ej(ω 0n+φ 0 ) ). 1 α (34) Then, u 3 (n) u 4 (n) A (1 α) Im( jn s0 (n)e j(ω 0n+φ 0 ) ) A (1 α) n s0 (n) (35) Assuming α is close to unity or the SNR is sufficient large, it sts that u 3(n) u 4 (n) A (1 α) n s0 (n) 1. Thus, u 4 (n) in Eq. 7 can be neglected. Therefore, by subtracting ω 0 from both sides of Eq. 7 assuming u(n) u 1 (n) + u (n) β 1 μa /(1 α),weobtain θ(n + 1) β θ(n) + μu(n). (36) With Eq. 36, the transform function from u(n) to θ(n) is written as: H u θ (z) μz 1. (37) 1 βz 1 Hence, the MSE of the estimated frequency can be expressed as [6]: E { θ(n) } r θ (0) Hu θ (z)h ( ) 1 u θ z Ru (z)z 1 dz, (38) πj R u (z) denotes the z-transform of r u (l), which is the autocorrelation sequence of u(n) can be calculated as: r u (l) E{u(k + l)u (k)} r u1 (l) + r u (l) + r u1 u (l), (39) r u1 (l) E[ u 1 (n + l)u 1 (n)], (40)

5 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page 5 of 11 r u (l) E[ u (n + l)u (n)], (41) r u1 u (l) E[ u 1 (n + l)u (n)]. (4) Thus R u (z) in Eq. 38 can be divided into three parts: R u (z) Z{r u (l)} R u1 (z) + R u (z) + R u1 u (z), (43) R u1 (z), R u (z),r u1 u (z) denote the z-transform of r u1 (l), r u (l), r u1 u (l), which will be calculated in what follows. To get r u1 (l),wetransformeq.9as: u 1 (n) s s 0 (n)n s1 (n) s s0 (n)n s1 (n), (44) j then Eq. 40 can be rearranged as: r u1 (l) E[ u 1 (n + l)u 1 (n)] p i (l), (45) p 1 (l) E { s s0 (n + l)n s1 (n + l)s s0 (n)n s1 (n) }, (46) i1 p (l) E { s s0 (n + l)n s1 (n + l)s s0 (n)n s1 (n) }, (47) p 3 (l) E { s s0 (n + l)n s1 (n + l) s s0 (n)n s1 (n) }, (48) p 4 (l) E { s s0 (n + l)n s1 (n + l) s s0 (n)n s1 (n) }. (49) By using the results in Appendix A considering that s s0 (n) n s1 (n) are uncorrelated, we can rewrite Eqs. 46, 47, 48, 49 as p 1 (l) ζ ss0 s s0 (l)ζ ns1 n s1 (l) 0, (50) p (l) ζ ss0 s s0 (l)ζ ns1 n s1 (l) 0, (51) p 3 (l) r ss0 (l)r ns1 ( l), (5) p 4 (l) r ss0 ( l)r ns1 (l), (53) r ss0 (l) E{s s0 (n)s s0 (n l)}, (54) r ns1 (l) E{n s1 (n)n s1 (n l)}. (55) Substituting Eqs. 50, 51, 5, 53 into Eq. 45, we get r u1 (l) 1 [ ] r ss0 (l)r ns1 ( l) + r ss0 ( l)r ns1 (l). (56) 4 Considering Eq. 6, r ss0 (l) in Eq. 56 can be written as: r ss0 (l) E{s s0 (n)s s0 (n l)} A e jω 0l (1 α). (57) Substituting Eq. 57 into Eq. 56 yields [ ] A r ns1 ( l)e jω0l + r ns1 (l)e jω 0l r u1 (l) 4(1 α). (58) The z-transform of both sides of Eq. 58 can be expressed as: [ A ( R ns1 z 1 e jω ) ( ) ] 0 + R ns1 ze jω 0 R u1 (z) 4(1 α). (59) Note that R ns1 (z) can be exped as [6]: R ns1 (z) H s1 (z)h s1 (1/z )R n (z), (60) R n (z) Z{v(n)} σ v H s 1 (z) 1+k 0z 1 1+αk 0 z 1.Utilizing the Taylor series expansion e j θ 1+j θ +o( θ ), we obtain A σv (z 1)(1 z) R u1 (z) (1 α) (61) (z α)(1 αz) Using the similar method of deriving R u1 (z), wegetthe following results (see Appendix B for details) R u (z) σ 4 v (1 α ), (6) R u1 u (z) 0. (63) Substituting Eqs. 61, 6, 63 into Eq. 38, finally we get [ ] E { θ(n) } μ A σv 1 αβ + σ v 4(1 α) (1 β) (1 + β)(1 + α)(1 α). (64) Equation 64 indicates that the estimated MSE is independent of input frequency ω 0 smooth factor ρ. 5 Simulation results Computer simulations are conducted to confirm the effectiveness of the proposed algorithm the validity of the theoretical analysis results. 5.1 Performance comparisons In the following two simulations, the proposed algorithm is compared with four conventional algorithms [14, 16, 17, 19] under two different kinds of inputs, namely a

6 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page 6 of 11 fixed frequency input a quadratic chirp input. The input signal takes the form x(n) e j(ϕ(n)+θ0) + v(n), ϕ(n) is the instantaneous phase. The parameters are adjusted to establish an equal steady-state MSE an equal notch bwidth for all the algorithms. The initial notch frequency value is set to zero for all the methods. Figure presents the MSE curves of five algorithms with a fixed frequency ϕ(n) 0.4πn at SNR 10 0 db, respectively. Note that the proposed algorithm outperforms the other four algorithms. The NCPG algorithm achieves the similar convergence rate as the proposed algorithm at SNR 10 db while the former diverges at SNR 0 db. This indicates that the proposed algorithm is robust even at very low SNR conditions. Figure 3 presents the tracking rate of the five algorithms with a quadratic chirp input signal: ϕ(n) A c (φ 1 n + φ n + φ 3 n 3 ),φ 1 π/4, φ π/ 10 3 φ 3 π/ Parameter A c is adopted to control the value of chirp rate. For this case, the desired true frequency can be obtained by ϕ(n)/ n A c (φ 1 + φ n + 3φ 3 n). Figure 3a depicts the tracking MSE obtained when A c 1, Fig. 3b presents the MSE with an increased chirp rate: A c. The results imply that under the non-stationary case, the proposed method can achieve faster convergence speed than all the other four algorithms. When tracking speed is concerned, we see that the RLS-SM method the proposed method can maintain an equally small MSE than the other three methods especially at the high chirp rate part. We checked each of the learning curves of the NCPG algorithm found that this algorithm even diverges in some runs. 5. Simulations of steady-state estimation MSE In the following four simulations, the simulated steadystate MSE of the proposed algorithm is compared with the theoretical results in Eq. 64 with different input frequency ω 0, SNR, pole radius α step size μ. Thesimulation results are obtained by averaging over 500 trials. Figure 4 displays the comparison of the theoretical simulated steady-state MSEs versus signal frequency ω 0 under two different SNRs (SNR db). The curves show that the theoretical MSEs can predict the simulated MSEs precisely, the steady-state MSEs are independent of input frequency ω 0.Wealsoseethata higher SNR leads to a larger MSE. Figure 5 exhibits the comparison of the theoretical simulated steady-state MSEs versus SNR under two different parameter settings: (1) α 0.9,μ 0.8 () α 0.98, μ 0.1. The proposed approach predicts the MSEs well, although some discrepancies are observed with α 0.9, μ 0.8. That is because the CANF can hardly converge when the SNR is very low. (a) 0 MSE (db) 0 40 Proposed (μ 0.1) LCANF [17] (μ 0.003) MCPG [16] (μ 0.009) RLS SM [14] (κ 0.) NCPG [19] (μ 0.1) (b) MSE (db) Proposed (μ 0.1) LCANF [17] (μ 0.003) MCPG [16] (μ 0.005) RLS SM [14] (κ 0.) NCPG [19] (μ 0.1) Iteration Number Fig. Comparison of the convergence rates of the estimated MSE under two different SNRs (α 0.9,ρ 0.8, 1000 runs): a SNR 10 db b SNR 0 db

7 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page 7 of 11 Fig. 3 Comparison of the tracking behaviors for a quadratic chirp input under two different chirp rates (α 0.9, SNR 0 db, 1000 runs): a comparison of MSEs when A c 1b comparison of MSEs when A c Figure 6 illustrates the comparison of the theoretical simulated steady-state MSEs versus pole radius α.whenα decreases, the MSEs increase the mismatch between the theoretical simulated steady-state MSEs is somewhat large. It is because Eq. 36 is derived on the basis of the assumption that α is close to unity. When α is small, the assumption does not hold. This explains the mismatch in Fig. 6. This finding implies that the theoretical MSE remains valid when α is close to unity. As shown in Fig. 7, the theoretical MSEs can predict the simulated steady-state MSEs well particularly for μ< 1.8 but the mismatch occurs when μ approaches the up MSE (db) SNR 60 db (Simulation1) SNR 60 db (Theory1) SNR 10 db (Simulation) SNR 10 db (Theory) Normalized Frequency 1 Fig. 4 Comparison of the theoretical simulated steady-state MSEs versus signal frequency ω 0 at SNR 60 db 10 db (α 0.9 μ 0.8) Fig. 5 Comparison of the theoretical simulated steady-state MSEs versus SNR (ω 0 0.π): (1) α 0.9, μ 0.8 () α 0.98, μ 0.1

8 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page 8 of 11 in the low SNR conditions () theoretical analysis of the proposed algorithm is in good agreement with computer simulation results. By cascading the proposed first-order gradient-adaptive lattice filters, the algorithm can be extended to hle complex signal with multiple sinusoids, which will be the focus of our further research. Appendix A Given complex sequences f (n) g(n), wedefineanew function ζ fg (l) as ζ fg (l) E { f (n + l)g(n) }. (65) Thus, for the input signal x(n) defined in Eq. 1, we have Fig. 6 Comparison of the theoretical simulated steady-state MSEs versus pole radius α (ω 0 0.π, μ 0.1, 500 runs): (1) SNR 60 db () SNR 10 db ζ xx (l) E {x(n + l)x(n)} { } A e jω0l E e j(ω 0n+φ 0 ) + ζ vv (l). (66) boundary of the step size. Moreover, it is noted that a large step size yields a large MSE. 6 Conclusions This paper has presented a complex adaptive notch filter based on the gradient-adaptive lattice approach. The new algorithm is computationally efficient can provide an unbiased estimation. The closed-form expressions for the steady-state MSE the upper bound of step size have been worked out. Simulation results demonstrate that (1) the proposed algorithm can achieve faster convergence rate than the traditional methods particularly Given that φ 0 is uniformly distributed over [ 0, π),we have E{e j(ω 0n+φ 0 ) }0. v(n) v r (n) + jv i (n) is assumed to be a zero-mean white complex Gaussian noise process v r (n) v i (n) are uncorrelated zero-mean real white noise processes with identical variances. Therefore, we have the following relations: r vr (l) σ v δ(l), (67) r vi (l) σ v δ(l), (68) r vr v i (l) r vi v r (l) 0, (69) r vr (l) r vi (l) are the autocorrelation sequences of v r (n) v i (n), respectively. r vr v i (l) is the crosscorrelation sequence of v r (n) v i (n). Consequently, we obtain ζ vv (l) E[ v(n + l)v(n)] r vr (l) r vi (l) + jr vr v i (l) 0. (70) Substituting Eq. 70 into Eq. 66, we get Fig. 7 Comparison of the theoretical simulated steady-state MSEs versus step size μ.(ω 0 0.π, α 0.95, 500 runs): (1) SNR 60 db () SNR 10 db ζ xx (l) 0. (71) Suppose y(n) h(n) x(n), h(n) denotes the impulse response of an arbitrary linear system. Then,

9 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page 9 of 11 ζ xy (l) E[ x(n + l)y(n)] E x(n + l) h(k)x(n k) k k k h(k)ζ xx (l + k) h( k)ζ xx (l k) h(l) ζ xx (l). (7) Moreover, ζ yy (l) E[ y(n + l)y(n)] E y(n + l) h(k)x(n k) k k k h(k)e [ y(n + l)x(n k) ] h(k)ζ xy ( l k). (73) Substituting Eq. 7 into Eq. 73 considering Eq. 71, we get ζ yy (l) h( l) h(l) ζ xx (l) 0 (74) By using Eq. 74, it is clear that E{y(n) }ζ yy (0) 0. (75) Appendix B To get r u (l),wetransformeq.30as: u (n) n s 0 (n)n s1 (n) n s0 (n)n s1 (n), (76) j then Eq. 41 can be rearranged as: r u (l) E[ u (n + l)u (n)] 1 4 q i (l), (77) 4 i1 q 1 (l) E{n s0 (n + l)n s1 (n + l)n s0 (n)n s1 (n)}, (78) q (l) E { n s0 (n + l)n s1 (n + l)n s0 (n)n s1 (n) }, (79) By assuming that n s0 (n) n s1 (n) are jointly Gaussian stationary processes utilising the Gaussian moment factoring theorem [7], we get q 1 (l) cum(n s0 (n + l), n s1 (n + l), n s0 (n), n s1 (n)) + r ns1 n s0 ( l)r ns1 n s0 (l) +r ns1 n s0 (0)r ns1 n s0 (0) + ζ n s0 n (l)ζ s ns1 0 n s1 (l), (8) cum( ) denotes high order cumulants of the complex rom variables. We adopt the widely used independence assumption [8], which tells that the present sample is independent of the past samples. Thus, we have cum(n s0 (n + l), n s1 (n + l), n s0 (n), n s1 (n)) 0. And, furthermore, considering ζ n s0 n s 0 (l) ζ ns1 n s1 (l) are all zero (see Appendix A), q 1 (l) in Eq. 8 can be rewritten as q 1 (l) r ns1 n s0 (0)r ns1 n s0 (0) +r ns1 n s0 ( l)r ns1 n s0 (l), (83) r nsi n sj (l) E{n si (n)n sj (n l)}, fori j. Utilizing the same method, we get q (l) r ns0 n s1 (0)r ns0 n s1 (0) +r ns0 n s1 ( l)r ns0 n s1 (l), (84) q 3 (l) r ns0 ( l)r ns1 (l) r ns1 n s0 (0)r ns0 n s1 (0), (85) q 4 (l) r ns0 (l)r ns1 ( l) r ns1 n s0 (0)r ns0 n s1 (0), (86) r nsi (l) E{n si (n)n si (n l)}, i {0, 1}, Substituting Eqs. 83, 84, into Eq. 77, we get r u (l) 1 [ r ns1 n 4 s0 (0) + r ns0 n s1 (0) r ns1 n s0 (0)r ns0 n s1 (0) r ns0 (l)r ns1 ( l) + r ns0 n s1 ( l)r ns0 n s1 (l) r ns0 ( l)r ns1 (l) ] + r ns1 n s0 ( l)r ns1 n s0 (l). (87) In the following part, the exact forms of r ns1 (l), r ns0 (l), r ns1 n s0 (l), r ns0 n s1 (l) are derived. Note that R ns1 (z) can be exped as [6] R ns1 (z) H s1 (z)h s1 (1/z )R n (z) ( 1 + k0 z 1)( 1 + k0 z) q 3 (l) E { n s0 (n + l)n s1 (n + l) n s0 (n)n s1 (n) }, (80) q 4 (l) E { n s0 (n + l)n s1 (n + l) n s0 (n)n s1 (n) }. (81) σv ( 1 + αk0 z 1)( 1 + αk0 [ z) σ v (1 α) 1 (1 + α)α 1 + (αk0 ) 1 z α 1 α αk 0 z 1 ], (88)

10 Zhu et al. EURASIP Journal on Advances in Signal Processing (016) 016:79 Page 10 of 11 R n (z) σv H s 1 (z) 1+k 0z 1.Sincer 1+αk 0 z 1 ns1 (l) is a two-sided sequence with the region of convergence given by k 0 /α > z >α k 0, the inverse z-transform of R ns1 (z) can be expressed as r ns1 (l) 1 πj σ v (1 α) (1 + α)α R ns1 (z)z l 1 dz [ ( (αk 0 ) 1) l u( l 1) α 1 α δ(l) ( αk 0) l u(l)], (89) u(l) denotes the unit step sequence. Using the same method, we have r ns0 (l) [ σ v α ( (αk0 1 ) 1) l u( l 1) ] ( αk 0 ) l u(l), (90) r ns1 n s0 (l) σ v [ ( (αk 1 + α 0 ) 1) l u( l 1) α α δ(l) 1 ] α ( αk 0) l u(l), (91) r ns0 n s1 (l) σ v [ α α ( ( αk0 ) 1) l u( l 1) +( αk 0 ) l u(l)]. (9) Substituting Eqs. 89, 90, 91, 9 into 87, taking the z-transform on both sides we have R u (z) Z{r u (l)} σ 4 v (1 α ). (93) Substituting Eqs into Eq. 4 considering that s s0 (n) is uncorrelated with n s1 (n) n s0 (n),wehave r u1 u (l) 1 4 E{s s 0 (n + l)}e { n s1 (n + l) [ n s0 (n)n s1 (n) n s0 (n)n s1 (n) ]} 1 4 E{s s 0 (n + l)}e { n s1 (n + l) [ n s0 (n)n s1 (n) n s0 (n)n s1 (n)] }. (94) Since s s0 (n) is a zero-mean stationary process, it holds that r u1 u (l)0. Thus we get R u1 u (z) Z{r u1 u (l)} 0. (95) Acknowledgements This work is supported by Strategic Priority Research Program of the Chinese Academy of Sciences under Grants XDA , in part by the National Science Fund of China under Grant We thank the reviewers for their constructive comments suggestions. Competing interests The authors declare that they have no competing interests. Received: 9 January 016 Accepted: 9 June 016 References 1. L-M Li, LB Milstein, Rejection of pulsed cw interference in pn spread-spectrum systems using complex adaptive filters. IEEE Trans. Comm. COM-31, 10 0 (1983). 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