Mathematical Problems in Engineering

Volume 2018, Article ID 3490830, 7 pages

https://doi.org/10.1155/2018/3490830

## A Novel Design of Sparse FIR Multiple Notch Filters with Tunable Notch Frequencies

^{1}School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China^{2}Tianjin Key Laboratory of Optoelectronic Detection Technology and Systems, Tianjin 300387, China^{3}College of Electronic Information and Optical Engineering, Nankai University, Tianjin 300071, China

Correspondence should be addressed to Jiaxiang Zhao; nc.ude.iaknan@xjoahz

Received 27 December 2017; Revised 14 March 2018; Accepted 2 April 2018; Published 8 May 2018

Academic Editor: Eric Feulvarch

Copyright © 2018 Wei Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We focus on the design of finite impulse response (FIR) multiple notch filters. To reduce the computational complexity and hardware implementation complexity, a novel algorithm is developed based on the mixture of the tuning of notch frequencies and the sparsity of filter coefficients. The proposed design procedure can be carried out as follow: first, since sparse FIR filters have lower implementation complexity than full filters, a sparse linear phase FIR single notch filter with the given rejection bandwidth and passband attenuation is designed. Second, a tuning procedure is applied to the computed sparse filter to produce the desired sparse linear phase FIR multiple notch filter. When the notch frequencies are varied, the same tuning procedure can be employed to render the new multiple notch filter instead of designing the filter from scratch. The effectiveness of the proposed algorithm is demonstrated through three design examples.

#### 1. Introduction

The multiple notch filters, which can highly attenuate some frequency components in the input signal while leaving the others relatively unchanged, are widely used in many applications. Important examples include radar systems, control and instrumentation systems, communications systems, medical applications, biomedical engineering, and indoor localization [1, 2].

Various methods [3–8] have been reported to design FIR multiple notch filters. In general, the multiple notch filters derived from these algorithms are not sparse. Compared with full FIR filters, sparse filters can significantly reduce the implementation complexity in the hardware. In [9], we proposed an iterative reweighed OMP algorithm to compute sparse notch filters. However, when the notch frequencies are varied, it requires one to design the whole filter from scratch, hence increasing the computational complexity of this scheme.

Recently, in [10–12], a number of algorithms are proposed to design FIR filters based on LMS minimization or Monte Carlo methods. The disadvantage of these approaches is the suboptimality in terms of the filter length related to its selectivity. Another disadvantage is that the attenuation at the notch frequency changes during the adaptation process; therefore, a strong attenuation of the disturbing signal at the notch frequency is not guaranteed. Moreover, the actual value of the attenuation at notch frequency is caused by the adaptation process.

In this brief, the design problems of sparse FIR multiple notch filters with tunable notch frequencies are studied. To reduce the computational complexity and the hardware complexity, a novel algorithm is developed based on the mixture of the tuning of notch frequencies and the sparsity of filter coefficients. The sparse FIR multiple notch filters can significantly reduce the number of the adders and multipliers used in the hardware implementation. However, the design of FIR sparse filter always involves iterative procedures and numerical optimization, which results in a high computational complexity for the practice system. The tuning of notch frequencies is a useful operation for the design of FIR multiple notch filter. In the case of variable notch frequencies, the same tuning process is implemented to obtain the new multiple notch filter instead of designing the filter from scratch. Therefore, the tuning feature can significantly reduce the computational complexity. We demonstrate the effectiveness of this approach through three design examples.

#### 2. Problem Formulation

Given the design parameters of linear phase FIR multiple notch filter, which include a set of the notch frequencies , rejection bandwidth , and passbands attenuation , the given notch frequencies satisfying for are allowed to be nonuniformly distributed in the set . The ideal multiple notch filter amplitude response satisfieswhere and are, respectively, defined as

To simplify the presentation, we focus on the design of Type-I linear phase FIR filter ; that is, the filter order is even and for all . For other types of filter, our design method presented in this letter is feasible. For the case of Type-I filter, the zero-phase amplitude response can be expressed aswith .

#### 3. The Proposed Sparse Linear Phase FIR Multiple Notch Filter Design

In this section, a novel design method is presented to produce the sparse FIR multiple notch filter. The procedure of computing the linear phase FIR multiple notch filter starts with the estimation of the initial order of the filter throughFrom [13, eq. ], is computed aswhere and the function is determined by [13, eq. ]. The arguments of can be computed as

The following design procedure is mainly comprised of two stages: in the first stage, a sparse linear phase FIR single notch filter with the given rejection bandwidth and passband attenuation is designed as a fixed sparse filter. In the next stage, a tuning process is carried out to compute the desired multiple notch filter with the given notch frequencies based on the filter .

##### 3.1. Sparse Linear Phase FIR Single Notch Filter Design

In this section, a sparse linear phase FIR single notch filter of order with the notch frequency is designed. Let represent the single notch filter, as shown in Figure 1; the real-valued amplitude response satisfies