A generalized design framework for IIR digital multiple notch filters
- Qiusheng Wang^{1}Email author and
- Deepa Kundur^{2}
https://doi.org/10.1186/s13634-015-0210-5
© Wang and Kundur; licensee Springer. 2015
Received: 15 August 2014
Accepted: 25 February 2015
Published: 20 March 2015
Abstract
Digital multiple notch filters are used in a variety of applications to remove or suppress multiple sinusoidal or narrow-band interference in digital signals. In this paper, we propose an all-pass filter-based design framework for infinite impulse response (IIR) multiple notch filters. Our approach aims to overcome the limitations of former techniques through greater design capacity and performance. The proposed framework has versatility and enables the tailored use of design constraints thus providing a family of possible multiple notch filter design methods. The design performance and practicality of the proposed framework are verified empirically by a series of experimental results and different applications.
Keywords
Multiple notch filter Digital notch filter Digital all-pass filter Infinite impulse response Digital filter design Filter design framework Weighted least square Digital signal processing1 Introduction
Digital notch filters are used to eliminate sinusoidal or narrow-band interference in digital signals while preserving the other frequency components intact [1-4]. They have been widely applied in a variety of disciplines ranging from audio signal processing to biomedical engineering [5-19]. Recently, there has been interest in designing multiple-frequency notch filters that aim to suppress two or more frequencies of an input signal [20-25]. Two types, finite-impulse response (FIR) and infinite-impulse response (IIR), exist. As typical, the former exhibits advantages of linear phase and stability while the latter has advantages of narrower stop-band and higher quality factor for the same filter order of more significance to notch filter design [26-34].
Several approaches exist to design IIR multiple notch filters [1-3,21-23,35-40]. The cascading method is a popular direct design technique that realizes a multiple notch filter by cascading several well-designed second-order IIR single-notch filters [36]. The resulting system is characterized by a canonical structure for software and hardware realization. The main drawback is that it is restricted to few notch frequencies and very narrow bandwidths otherwise leading to non-unity pass-band gains [37].
The optimal design method requires computations for pole placement that results in a better frequency characteristic [2,3,37,38], but highly complex search-based or iterative algorithms are required even for a notch filter with two frequencies [2,38]. For a greater number of notch frequencies, a two-stage approach has been proposed to generate a stable optimal multiple notch filter, but the design procedure involves quadratic programming via an iterative scheme [3].
The all-pass filter-based method presented by S-C. Pei and C-C. Tseng transforms the specifications of a multiple notch filter to those of an equivalent all-pass filter. A linear design equation is constructed to determine the all-pass filter coefficients which are used to characterize the desired multiple notch filters [1,41]. An improved method was proposed to design a notch filter with only two frequencies [42]. One advantage is that all-pass filter-based methods have normalized analytical forms with no iterative calculations.
The practical success of all-pass filter-based methods is a result of three assumptions: i) the magnitude responses are symmetric about notch frequencies, ii) the neglected right-hand cutoff frequencies do not produce design errors, and iii) the constructed linear equation has an accurate solution. However, these constraints can also lead to the following challenges [42]. The symmetric magnitude responses are too prohibitive for applications. The information of the all-pass filter is not fully utilized. And, the linear equation is potentially ill-conditioned due to tangent operations.
Thus in this work, we present a generalized design framework for IIR multiple notch filters whereby we aim to: a) deduce an improved design process of the all-pass filter-based method, b) maximize the availability of phase information of all-pass filter, and c) introduce selection mechanisms and weighting strategies to enable the unified representation of a family of all-pass filter-based methods. We also apply the framework to typical multiple notch filter applications to empirically test performance.
In the next section, an improved all-pass filter-based approach is explored for multiple notch filters. Section 2 proposes a generalized framework with selection mechanisms and weighting strategies. Section 2 verifies the effectiveness and practicality of our framework through experimental results. Final conclusions are drawn in Section 2.
2 Improved multiple notch filter design
where A _{ i } and ϕ _{ i } represent the magnitude and initial phase of the ith sinusoidal component.
where i=1,2,⋯,N. Without loss of generality, we assume ω _{ N1}<ω _{ N2}<⋯<ω _{ NN }. In practice, zero bandwidths can not be realized [22]. Hence, the frequency response of an actual notch filter is the approximation to that of the ideal one.
2.1 All-pass filter-based design process
where A(z) is an 2N-order all-pass filter.
Therefore, θ _{ A }(ω) must be carefully selected such that H(e ^{ j ω }) fulfills the design specifications (notch frequencies and bandwidths) of the notch filter.
where real coefficients a _{ k }∈ℝ (k=1,2,⋯,2N) in both the numerator and denominator polynomials [1,3]. Given the formulation in Equation 4, determining appropriate values of a _{1},a _{2},⋯,a _{2N } represents the core problem of all-pass filter-based multiple notch filter design.
Thus to solve for a _{1},a _{2},⋯,a _{2N }, at least 2N distinct pairs (ω,θ _{ A }(ω)) must be known to provide at least 2N equations to solve 2N unknowns.
It is well known that a unique solution to Equation 16 exists if M=2N and Q is non-singular. For M>2N, Equation 16 only has a least-square solution [44].
- 1.
Choose the related all-pass filter design constraints {(ω _{ m },θ _{ A }(ω _{ m }))|_{ m=1,2,⋯,M }} to construct Equation 16 where M≥2N.
- 2.
Solve the linear equations of Equation 16 to obtain the all-pass filter coefficient vector a=[a _{1},a _{2},⋯,a _{2N }]^{ T }.
- 3.
Substitute a _{1},a _{2},⋯,a _{2N } into Equation 6 and then Equation 3 to obtain the multiple notch filter system function H(z).
We assert that step 1 will have a significant impact on the quality of the multiple notch filter design. Thus, in the next section, we assess four strategies to select design constraints that we verify and compare empirically in Section 2.
2.2 Constraint selections of notch filter design
For a 2N-order all-pass filter, its phase response θ _{ A }(ω) is monotonically decreasing from 0 to −2N π as illustrated in Figure 1. Given the continuity of both ω and θ _{ A }(ω), there are an uncountably infinite number of (ω,θ(ω)) pairs that can be selected for notch filter design by Equation 16. However, if we restrict ourselves to those quantities related to the design specifications of the multiple notch filter, we can naturally arrive at 3N pairs that may be used. Specifically, those pairs related to the notch frequencies and left-hand and right-hand 3 dB cut-off frequencies employed [1].
- 1.At notch frequencies, θ _{ A }(ω) must be equal to an odd multiple of π to ensure that |H(e ^{ j ω })|=0. Specifically, we require that θ _{ A }(ω _{ Ni })=−(2i−1)π,(i=1,2,⋯,N), to give \(A\left (e^{j\omega _{\textit {Ni}}}\right)= e^{j\theta _{A}(\omega _{\textit {Ni}})}=-1\) resulting in \(|H\left (e^{j\omega _{\textit {Ni}}}\right)|=0\). Therefore, {(ω _{ Ni },θ _{ A }(ω _{ Ni }))|_{ i=1,2,⋯,N }} gives \(\phantom {\dot {i}\!}|H(e^{j\omega _{\textit {Ni}}})|=0\) at the N notch frequencies providing a valuable set of filter design constraints.
- 2.
Next, at the left-hand 3 dB frequencies, θ _{ A }(ω) must appropriately ensure that \(|H\left (e^{j\omega }\right)|=\sqrt {2}/2\). One way to achieve this is to let θ _{ A }(ω _{ Li })=−(2i−1)π+π/2,(i=1,2,⋯,N) such that \(A\left (e^{j\omega _{\textit {Li}}}\right)= e^{j\theta _{A}(\omega _{\textit {Li}})}=-j\) resulting in \(|H\left (e^{j\omega _{\textit {Li}}}\right)|=\sqrt {2}/2\). Thus, {(ω _{ Li },θ _{ A }(ω _{ Li }))|i=1,2,⋯,N} gives \(|H\left (e^{j\omega _{\textit {Li}}}\right)|=\sqrt {2}/2\) providing another valuable set of filter design constraints.
- 3.
Similarly, the right-hand 3 dB frequencies with θ _{ A }(ω _{ Ri })=−(2i−1)π−π/2,(i=1,2,⋯,N) give \(A\left (e^{j\omega _{\textit {Ri}}}\right)= e^{j\theta _{A}(\omega _{\textit {Ri}})}=j\) resulting in \(|H\left (e^{j\omega _{\textit {Ri}}}\right)|=\sqrt {2}/2\) providing {(ω _{ Ri },θ _{ A }(ω _{ Ri }))|i=1,2,⋯,N} as the third set of filter design constraints.
Relationships among ω , θ _{ A } ( ω ), and | H ( e ^{ j ω } )|, for i =1,2,⋯, N
ω | θ _{ A } ( ω ) | | H ( e ^{ j ω } )| | |
---|---|---|---|
ω _{ Li } | ω _{ i }−B w _{ i }/2 | −(2i−1)π+π/2 | \(\sqrt {2}/2\) |
ω _{ Ni } | ω _{ i } | −(2i−1)π | 0 |
ω _{ Ri } | ω _{ i }+B w _{ i }/2 | −(2i−1)π−π/2 | \(\sqrt {2}/2\) |
Parameter constraint selection methods for Equation 16 , for i =1,2,⋯, N
Method I | Method II | Method III | Method IV | |
---|---|---|---|---|
(ω _{ Li },θ _{ A }(ω _{ Li })) | ∙ | ∙ | ∙ | |
(ω _{ Ni },θ _{ A }(ω _{ Ni })) | ∙ | ∙ | ∙ | |
(ω _{ Ri },θ _{ A }(ω _{ Ri })) | ∙ | ∙ | ∙ | |
Value of M | 2N | 2N | 2N | 3N |
The reader should note that each selection method from Table 2 provides sufficient information to solve Equation 16 since M≥2N, but they may lead to slightly different results for the same design specifications. We evaluate their design performance in Section 2 but characterize each approach briefly below.
Method I utilizes notch and left-hand cutoff frequencies as filter design constraints, which is suitable for the case of very narrow interference bandwidths and symmetrical magnitude responses about the notch frequencies. Method I is adopted in [1].
Method II, in contrast to method I, utilizes notch and right-hand cutoff frequencies for filter design. Roughly, it has the same applicability and limitations as method I, but the results of method II are slightly different from those of method I.
Method III employs left-hand and right-hand cutoff frequencies as the constraints for filter design. Although it is not limited to very narrow notch bandwidths and symmetrical magnitude responses, it may result in notch frequencies drifting from their desired positions.
Method IV makes use of all available information: notch frequencies and both left-hand and right-hand cutoff frequencies. Thus, it is able to realize a trade-off between notch bandwidths and notch positions effectively enhancing the design performance. The constraint selection method is also adopted by method V in Section 2.
The reader should note that M=2N in methods I, II, and III produce unique solutions. In contrast to method IV, M=3N results in an over-determined equation with a least-square solution. However, we aim to unify both into a generalized design framework that we present next.
3 Multiple notch filter design framework
3.1 Matrix representations of selection methods
where ω and \(\boldsymbol {\hat \omega }\) are 3N×1 vectors. \({\boldsymbol {\hat \omega }}_{L}\), \({\boldsymbol {\hat \omega }}_{N}\), and \({\boldsymbol {\hat \omega }}_{R}\) are N×1 vectors. S=diag(S _{ L },S _{ N },S _{ R }) is a 3N×3N diagonal matrix. S _{ L }, S _{ N }, and S _{ R } are N×N sub-matrices equal to either 0 or I such that S maps the set of all possible frequencies of interest ω to \(\boldsymbol {\hat \omega }\) employed for notch filter design.
Relationships between parameter constraint methods and selection matrix S
Method I | Method II | Method III | Method IV | |
---|---|---|---|---|
S _{ L } | I | 0 | I | I |
S _{ N } | I | I | 0 | I |
S _{ R } | 0 | I | I | I |
where θ and \({\boldsymbol {\hat {\theta }}}\) are 3N×1 vectors; \({\boldsymbol {\hat \theta }}_{L}\), \({\boldsymbol {\hat \theta }}_{N}\), and \({\boldsymbol {\hat \theta }}_{R}\) are N×1 vectors correspond to θ _{ L }, θ _{ N }, and θ _{ R } respectively; S has the same meaning as in Equation 19.
Therefore, four selection methods of (ω,θ _{ A }(ω)) in Table 2 can be uniformly represented via Equations 19 and 20 whereby the value of S determines the particular method used for multiple notch filter design.
3.2 Linear equation and least-square solution
where \({\boldsymbol {\hat {p}}}_{L}\), \({\boldsymbol {\hat {p}}}_{N}\), and \({\boldsymbol {\hat {p}}}_{R}\) are N×1 vectors; \({\boldsymbol {\hat {Q}}}_{L}\), \({\boldsymbol {\hat {Q}}}_{N}\), and \({\boldsymbol {\hat {Q}}}_{R}\) are N×2N sub-matrices. \(({\boldsymbol {{\hat p}}}_{L},{\boldsymbol {{\hat Q}}}_{L})\), \(({\boldsymbol {{\hat p}}}_{N},{\boldsymbol {{\hat Q}}}_{N})\), and \(({\boldsymbol {{\hat p}}}_{R},{\boldsymbol {{\hat Q}}}_{R})\) correspond to \(({\boldsymbol {\hat \omega }}_{L},{\boldsymbol {\hat \theta }}_{L})\), \(({\boldsymbol {\hat \omega }}_{N},{\boldsymbol {\hat \theta }}_{N})\), and \(({\boldsymbol {\hat \omega }}_{R},{\boldsymbol {\hat \theta }}_{R})\), respectively. Hence, \({\boldsymbol {\hat {p}}}\) is 3N×1 vector and \({\boldsymbol {\hat {Q}}}\) is 3N×2N matrix.
where \( {\widehat{\boldsymbol{Q}}}_I^{-1} \) is the inverse of \( {\widehat{\boldsymbol{Q}}}_I \). Similar forms and results as Equations 23 and 24 can be easily surmised for methods II and III.
Generally speaking, for methods I, II, and III, \(\boldsymbol {\hat {p}}\) and \(\boldsymbol {\hat {Q}}\) of Equations 21 degenerate to a 2N×1 vector and a 2N×2N nonsingular matrix, respectively. Because the rank of the resulting matrix \(\boldsymbol {\hat {Q}}\) is 2N, Equation 21 has a unique solution. In contrast, for method IV, \(\boldsymbol {\hat {p}}\) is a 3N×1 vector and \(\boldsymbol {\hat {Q}}\) a 3N×2N matrix whose rank is also 2N; thus, Equation 21 only has a least-square solution due to the structure and rank of \({\boldsymbol {\hat {Q}}}\). Thus, Equation 21 is a generalized form applicable to the various constraints discussed in Section 2.
3.3 Weighted design equation and its solution
For certain applications, the design precision requirements of notch frequencies are more rigorous than those at the cutoffs and vice versa. For example, in a contaminated electrocardiogram (ECG) signal, the interference resulting from fundamental and harmonic components of power lines have near-fixed frequency values [7,14,17]. Hence, the design precision requirement for the notch frequency values of the filter is stricter than those of the cutoff frequencies. The same is true of rejecting narrow-band interference from amplitude-modulated (AM) broadcasts in corona current of high-voltage direct current (HVDC) transmission lines.
Relationships between design methods and weighting matrix W
Method I | Method II | Method III | Method IV | |
---|---|---|---|---|
W _{ L } | I | 0 | I | I |
W _{ N } | I | I | 0 | I |
W _{ R } | 0 | I | I | I |
where \({\boldsymbol {\hat {W}}}=\text {diag}\left ({\boldsymbol {\hat {W}}}_{L},{\boldsymbol {\hat {W}}}_{N},{\boldsymbol {\hat {W}}}_{R}\right)\) is a 3N×3N diagonal matrix and \({\boldsymbol {\hat {W}}}_{L}={\bf {S}}_{L}{\bf {W}}_{L}\), \({\boldsymbol {\hat {W}}}_{N}={\bf {S}}_{N}{\bf {W}}_{N}\), and \({\boldsymbol {\hat {W}}}_{R}={\bf {S}}_{R}{\bf {W}}_{R}\) are N×N diagonal sub-matrices.
Relationships between design methods and weighting matrix \({\boldsymbol {\hat {W}}}\) , for α >1
Method I | Method II | Method III | Method IV | Method V | |
---|---|---|---|---|---|
\({\boldsymbol {{\hat {W}}}}_{L}\) | I | 0 | I | I | I |
\({\boldsymbol {{\hat {W}}}}_{N}\) | I | I | 0 | I | α I |
\({\boldsymbol {{\hat {W}}}}_{R}\) | 0 | I | I | I | I |
Thus, the incorporation of new design constraints can be easily achieved by modifying the weighting matrix of Equation 28. Method V in Table 5 is a new design method derived from Equation 29. It utilizes the same frequency selection method as method IV in Table 2. However, it treats notch and cutoff frequencies with different significance through the different weighting matrix. The advantages of method V is demonstrated in Section 2. The readers should note that Table 5 only provides five special cases of weighting matrix \({\boldsymbol {\hat {W}}}\). They can easily be modified to address different design requirements beyond those presented.
3.4 Generalized design framework
Our all-pass filter-based approach to multiple notch filter design can therefore be summarized as follows. Step 1 Acquire the design specifications of the multiple notch filter as a set of N notch frequencies and N notch bandwidths that form a set of 3N constraint frequencies presented in Table 1. Step 2 For the 3N frequencies in step 1, we can determine the 3N related all-pass filter phases from Table 1 and then produce 3N×1 vectors ω and θ. Step 3 Select one of the constraint methods from Table 2 and determine its corresponding selection matrix S from Table 3. From this, compute the frequency vector \(\boldsymbol {\hat \omega }\) via Equation 19 and the all-pass phase vector \(\boldsymbol {\hat \theta }\) from Equation 20. Step 4 Substitute all entries of \(\boldsymbol {\hat {\omega }}\) and \(\boldsymbol {\hat {\theta }}\) into Equations 17 and 18 to compute the 3N×1 vector \(\boldsymbol {\hat {p}}\) and 3N×2N matrix \(\boldsymbol {\hat {Q}}\), necessary in step 6. Step 5 Employing a measure of importance of notch, left-hand and right-hand cutoff frequencies determine a suitable 3N×3N weighing matrix W from Table 4 and then combine it with S to form \(\boldsymbol {\hat {W}}\) using Equation 28. Step 6 Construct the weighted equations \({\boldsymbol {\hat {W}}}{\boldsymbol {\hat {Q}}}{\bf {a}}={\boldsymbol {\hat {W}}}{\boldsymbol {\hat {p}}}\) and solve it by the least-squares technique of Equation 30. The solution a=[a _{1},a _{2},⋯,a _{2N }]^{ T } is the desired all-pass filter coefficient vector. Step 7 Substitute all entries of a into Equation 6 to gain system function A(z) of the all-pass filter. Then, substitute A(z) into H(z)=(1+A(z))/2 to obtain the system function H(z) of the desired multiple notch filter. Step 8 Let z=e ^{ j ω } and substitute it into H(z) to obtain H(e ^{ j ω }). Compare |H(e ^{ j ω })| with the given specifications to verify whether the design requirements are satisfied. If it is not, go to step 3 to choose a different method and/or step 6 to set a new weighting matrix and repeat the above design process.
where x[n] and y[n] are the input and output signals of the notch filter, respectively. Equation 32 is applied to perform notch filtering by iterative operations in time domain.
4 Experiments and results analysis
where ω∈[0,π]−[ω _{ Ni }−B _{ Ni }/2,ω _{ Ni }+B _{ Ni }/2], for i=1,2,⋯,N and H _{ d }(e ^{ j ω }) is defined by Equation 2.
4.1 Design performance among five methods
4.2 Comparisons with the classical methods
Figure 8 shows that the cascading method cannot meet the design requirements due to its uncontrollable gain. Figure 9 demonstrates that the optimal pole placement method and method V have similar magnitude responses, but the latter does not require iterative calculations. Figure 10 shows that the designed result of the method in [1] is the same as that of method I because they adopt the same all-pass filter phase information to design notch filter, but method I uses the degenerated design forms (shown in Equation 23) with no tangent computations. Therefore, method V has some advantages over those classical methods.
4.3 Interference elimination of ECG signals
It is worth noting that several oscillations occur near to the origin in Figure 11. This is due to edge effects (due to the iterative operations using Equation 32) at the beginning of the processing window and vanishes quickly.
4.4 Radio noise suppression of corona current
Based on the results above, we surmise the following: 1) irrespective of computational complexity and for the same specifications, method V obtains the best results among the derived approaches (methods I to V). 2) Method V outperforms the cascading method and the technique of [1] that exhibits the same results as method I. Moreover, method V has close results to the optimal pole placement technique but exhibits lower computational complexity. 3) The effectiveness and practicality of the proposed framework has been demonstrated by the experimental results on power line interference elimination of ECG signals and narrow-band radio noise suppression of HVDC corona currents.
4.5 Discussions
For a multiple digital filter with N notch frequencies, it has 3N constraints (N notch frequencies, N left-hand, and N right-hand cutoff frequencies). All-pass filter-based design methods attempt to find 2N appropriate coefficients to construct a 2N-order rational system function to approximate 3N constraints. The proposed framework enables a unified representation of the all-pass filter-based design methods using part or all (2N or 3N) constraints to find those 2N coefficients.
Among the derived methods from the proposed framework, methods I, II, and III only use 2N available phase information of all-pass filter. Their design equations are the special forms of the weighted design techniques. They are full rank and have exact solutions. Moreover, for the same design specifications, method I obtains the same results as the method in [1], but it avoids tangent computations which may result in ill-conditioned linear equations.
Methods IV and V utilize all 3N available phase information of all-pass filter. Their design equations are over-determined and only have least-square solutions. Method V achieves a better compromise between notch frequencies and cutoff frequencies. It obtains the best results among the derived methods, although it has very slight frequency drift which can be neglected in practice. The reader should note that in Section 2, the 2N notch filter coefficients of methods I, II, and III are directly solved from 2N×2N regular linear equations via Gaussian elimination, but those of methods IV and V are solved from 3N×2N over-determined linear equations using weighted least-square techniques. Hence, the computational complexity of methods IV and V are almost identical but higher than for methods I, II, and III.
In our design framework, the selection mechanisms provide the flexibility to choose a desired design method. The weighting strategies implement the compromise among the restricted conditions. Through the incorporation of diagonal selection matrix S and weighting matrix W, the improved diagonal matrix \({\boldsymbol {\hat {W}}}\) can implement selection and weighting simultaneously, but the calculation burden does not increase significantly. The computational efficiency of the proposed framework is far higher than that of the iteration-based design method.
5 Conclusions
In this paper, we present an all-pass filter-based design framework for digital multiple notch filters. The main contributions include: 1) the all-pass filter-based design process is reformulated as a novel linear equation which overcome the limits of classical design methods due to no tangent operations. 2) All-pass phase selection mechanisms and weighting strategies are introduced and incorporated to meet the diverse of requirements. They constitute the main parts of the presented framework under weighted least-square sense. Our framework enables the unified representation of a family of all-pass filter-based design methods. We highlight five design methods (methods I to V) derived from the presented framework. Their performance and effectiveness are compared and validated by a series of experimental results. Our investigations leads us to believe that method V has the best design performance and is recommended for use in practice.
Future work will extend our formation to: a) explore an adaptive IIR multiple notch filter design methodology and b) explore a FIR multiple notch filter by linear phase control mechanisms and weighting strategies.
Declarations
Acknowledgements
The authors would like to thank all the reviews for their helpful comments. The first author would like to thank the National Natural Science Foundation (60504023, 61273165) of China for funding this research.
Authors’ Affiliations
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