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# A gradient-adaptive lattice-based complex adaptive notch filter

- Rui Zhu
^{1}, - Feiran Yang
^{1}and - Jun Yang
^{1}Email author

**2016**:79

https://doi.org/10.1186/s13634-016-0377-4

© The Author(s) 2016

**Received:**9 January 2016**Accepted:**29 June 2016**Published:**16 July 2016

## Abstract

This paper presents a new complex adaptive notch filter to estimate and 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 and the upper bound of step size are also derived. Simulations are conducted to validate the theoretical analysis and demonstrate that the proposed method generates considerably better convergence rates and 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 and tracking technique that is utilised in a wide variety of applications, such as communication systems, biomedical engineering and radar systems [1–12]. The complex ANF (CANF) has recently gained much attention [13–20]. A direct-form poles and 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 and can result in biased estimations.

To address this problem, numerous efficient and 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], and 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 LMS-based 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 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] and the improved simplified lattice complex algorithm [20]. However, the former may be unstable in low signal-to-noise ratio (SNR) conditions, and 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 [21]. Instead of the traditional gradient estimation filter, we proposed a normalized lattice predictor that makes both forward and backward predictions. This scheme reduces computational complexity and enhances the robustness to noise influence. Furthermore, convergence rate is improved significantly when compared with conventional gradient-based or nongradient-based 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 and the steady-state mean square error (MSE) under different conditions. Computer simulations are conducted to confirm the validity of the theoretical analysis results and the effectiveness of the proposed algorithm.

The following notations are adopted throughout this paper. *j* denotes square root of minus one. ln[·] denotes the principal branch of the complex natural logarithm function and Im {·} means taking the imaginary part of a complex value. *Z*{·} and *E*{·} denote the z-transform operator and statistical expectation operator, respectively. *δ*(·) represents the Dirac function. Asterisk ∗ denotes a complex conjugate and ⊗ is the convolution operator.

## 2 Filter structure and adaptive algorithm

*x*(

*n*) with amplitude

*A*, frequency

*ω*

_{0}and initial phase

*ϕ*

_{0}:

where *ϕ*
_{0} is uniformly distributed over [0, 2*π*) and *v*(*n*)=*v*
_{
r
}(*n*)+*j*
*v*
_{
i
}(*n*) is assumed to be a zero-mean white complex Gaussian noise process. It is assumed *v*
_{
r
}(*n*) and *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}: \(H(z) = \frac {{1 - {e^{j\theta }}{z^{- 1}}}}{{1 - \alpha {e^{j\theta }}{z^{- 1}}}}\) where *θ* is the notch frequency and *α* represents the pole-zero constrained factor and determines the notch filter’s 3-dB attenuation bandwidth. The pole can remain in the unit circle by restricting the value of *α*.

*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*), where

*a*

_{0}is the coefficient of the all-pole filter. Then, a lattice predictor is employed to identify the forward and backward prediction errors

*s*

_{1}(

*n*) and

*r*

_{1}(

*n*), respectively. The transform functions from

*s*

_{1}(

*n*) and

*r*

_{1}(

*n*) to

*s*

_{0}(

*n*) are given by

*H*

_{ f }(

*z*)=

*N*(

*z*)=1+

*k*

_{0}

*z*

^{−1}and

*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:

Thus, *θ* can be computed as *θ*=Im{ln[−*k*
_{0}]}.

*k*

_{0}. We consider the following cost function:

*J*

_{ fb }with its instantaneous estimation, i.e.,

*θ*(

*n*), we obtain

*θ*(

*n*) is real, the adaptation equation can be written as

*μ*is the step size and the normalized signal

*ξ*(

*n*) can be recursively calculated as

where *ρ* denotes the smoothing factor.

## 3 Convergence analysis

*G*(

*θ*(

*τ*))=

*E*{

*s*

_{0}

^{∗}(

*n*)

*s*

_{0}(

*n*)} and

*S*

_{ x }(

*ω*) is the power spectral density (PSD) of

*x*(

*n*):

*S*

_{ x }(

*ω*)=2

*π*

*A*

^{2}

*δ*(

*ω*−

*ω*

_{0})+

*σ*

_{ v }

^{2}[17] and the transfer functions

*N*(

*e*

^{ j ω }) and 1/

*D*(

*e*

^{ j ω }) are defined in the previous section where

*e*

^{ j ω }is substituted by

*z*. Since Eq. 9 is the associated ordinary differential equation of the proposed adaptive algorithm, according to [23],

*θ*(

*n*) will always converge to the stationary point of Eq. 9 without exception, and this stationary point must satisfy \(\frac {d}{{d\tau }}\theta (\tau) = 0. \xi (\tau)\) 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*)]

^{2}.

*L*(

*τ*)≥0 for all

*τ*. Meanwhile,

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*) can always converge to the expected frequency *ω*
_{0} [23].

*μ*. Taking the expectation on both sides of Eq. 7, we obtain

*s*

_{0}(

*n*) is wide-sense stationary, we have

*Δ*

*ξ*(

*n*) is the zero-mean stochastic error sequence that is independent of the input signal. By applying Eq. 17 and disregarding the second-order error, we obtain

*μ*should satisfy:

Furthermore, when SNR→*∞* or *α*→1, we have *μ*∈(0,2], which is independent of the input.

## 4 Steady-state MSE analysis

*Δ*

*θ*(

*n*)=

*θ*(

*n*)−

*ω*

_{0}, we obtain the following two approximations: \(\mathop {\lim }\limits _{n \to \infty } \;{\text {sin}}(\Delta \theta (n)) \approx \Delta \theta (n)\) and \(\mathop {\lim }\limits _{n \to \infty } \;{\text {cos}}(\Delta \theta (n)) \approx 1.\) Then, the steady-state transfer function from

*s*

_{1}(

*n*) and

*s*

_{0}(

*n*) to

*x*(

*n*) can be written as:

*x*(

*n*) in Eq. 1 is assumed to be composed of a single frequency part and Gaussian white noise. Thus, the steady-state outputs

*s*

_{1}(

*n*) and

*s*

_{0}(

*n*) can be expressed as:

*s*

_{1}(

*n*) and

*s*

_{0}(

*n*), respectively. By using Eqs. 21 and 22, we obtain

Assuming *α* is close to unity or the SNR is sufficient large, it stands that \(\left | {\frac {{{u_{3}}(n)}}{{{u_{4}}(n)}}} \right | \ge \frac {A}{{(1 - \alpha)\left | {{n_{{s_{0}}}}^{*}(n)} \right |}} \gg 1\). Thus, *u*
_{4}(*n*) in Eq. 27 can be neglected.

*ω*

_{0}from both sides of Eq. 27 and assuming

*u*(

*n*)=

*u*

_{1}(

*n*)+

*u*

_{2}(

*n*) and \(\beta = 1 - \bar \mu {A^{2}}/{(1 - \alpha)^{2}}\), we obtain

*u*(

*n*) to

*Δ*

*θ*(

*n*) is written as:

*R*

_{ u }(

*z*) denotes the z-transform of

*r*

_{ u }(

*l*), which is the autocorrelation sequence of

*u*(

*n*) and can be calculated as:

*R*

_{ u }(

*z*) in Eq. 38 can be divided into three parts:

where \({R_{{u_{1}}}}(z)\), \({R_{{u_{2}}}}(z)\), and \({R_{{u_{1}}{u_{2}}}}(z)\) denote the z-transform of \({r_{{u_{1}}}}(l)\), \({r_{{u_{2}}}}(l)\), and \({r_{{u_{1}}{u_{2}}}}(l)\), which will be calculated in what follows.

*e*

^{ j Δ θ }=1+

*j*

*Δ*

*θ*+

*o*(

*Δ*

*θ*

^{2}), we obtain

Equation 64 indicates that the estimated MSE is independent of input frequency *ω*
_{0} and smooth factor *ρ*.

## 5 Simulation results

Computer simulations are conducted to confirm the effectiveness of the proposed algorithm and 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 fixed frequency input and a quadratic chirp input. The input signal takes the form \(\phantom {\dot {i}\!}x(n) = {e^{j(\varphi (n) + {\theta _{0}})}} + v(n)\), where *φ*(*n*) is the instantaneous phase. The parameters are adjusted to establish an equal steady-state MSE and an equal notch bandwidth for all the algorithms. The initial notch frequency value is set to zero for all the methods.

*φ*(

*n*)=0.4

*π*

*n*at SNR = 10 and 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.

*φ*(

*n*)=

*A*

_{ c }(

*ϕ*

_{1}

*n*+

*ϕ*

_{2}

*n*

^{2}+

*ϕ*

_{3}

*n*

^{3}), where

*ϕ*

_{1}=−

*π*/4,

*ϕ*

_{2}=

*π*/2×10

^{−3}and

*ϕ*

_{3}=−

*π*/6×10

^{−6}. 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}+2

*ϕ*

_{2}

*n*+3

*ϕ*

_{3}

*n*). Figure 3 a depicts the tracking MSE obtained when

*A*

_{ c }=1, and Fig. 3 b presents the MSE with an increased chirp rate:

*A*

_{ c }=2. 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 and 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 and found that this algorithm even diverges in some runs.

### 5.2 Simulations of steady-state estimation MSE

In the following four simulations, the simulated steady-state MSE of the proposed algorithm is compared with the theoretical results in Eq. 64 with different input frequency *ω*
_{0}, SNR, pole radius *α* and step size *μ*. The simulation results are obtained by averaging over 500 trials.

*ω*

_{0}under two different SNRs (SNR = 60 and 10 dB). The curves show that the theoretical MSEs can predict the simulated MSEs precisely, and the steady-state MSEs are independent of input frequency

*ω*

_{0}. We also see that a higher SNR leads to a larger MSE.

*α*=0.9,

*μ*=0.8 and (2)

*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.

*α*. When

*α*decreases, the MSEs increase and the mismatch between the theoretical and 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.

*μ*<1.8 but the mismatch occurs when

*μ*approaches the up 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 and can provide an unbiased estimation. The closed-form expressions for the steady-state MSE and 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 in the low SNR conditions and (2) 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 handle complex signal with multiple sinusoids, which will be the focus of our further research.

## 7 Appendix A

*f*(

*n*) and

*g*(

*n*), we define a new function

*ζ*

_{ fg }(

*l*) as

*x*(

*n*) defined in Eq. 1, we have

*ϕ*

_{0}is uniformly distributed over [0, 2

*π*), we have \(\phantom {\dot {i}\!}E\{ {e^{j2({\omega _{0}}n + {\phi _{0}})}}\} = 0\).

*v*(

*n*)=

*v*

_{ r }(

*n*)+

*j*

*v*

_{ i }(

*n*) is assumed to be a zero-mean white complex Gaussian noise process where

*v*

_{ r }(

*n*) and

*v*

_{ i }(

*n*) are uncorrelated zero-mean real white noise processes with identical variances. Therefore, we have the following relations:

*v*

_{ r }(

*n*) and

*v*

_{ i }(

*n*), respectively. \({r_{{v_{r}}{v_{i}}}}(l)\) is the cross-correlation sequence of

*v*

_{ r }(

*n*) and

*v*

_{ i }(

*n*). Consequently, we obtain

*y*(

*n*)=

*h*(

*n*)⊗

*x*(

*n*), where

*h*(

*n*) denotes the impulse response of an arbitrary linear system. Then,

## 8 Appendix B

*q*

_{1}(

*l*) in Eq. 82 can be rewritten as

*k*

_{0}|/

*α*>|

*z*|>

*α*|

*k*

_{0}|, the inverse z-transform of \({R_{{n_{{s_{1}}}}}}(z)\) can be expressed as

*u*(

*l*) denotes the unit step sequence. Using the same method, we have

## Declarations

### Acknowledgements

This work is supported by Strategic Priority Research Program of the Chinese Academy of Sciences under Grants XDA06040501, and in part by the National Science Fund of China under Grant 61501449. We thank the reviewers for their constructive comments and suggestions.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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