Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 936958, 14 pages

http://dx.doi.org/10.1155/2015/936958

## Minimum Phase Property of Chebyshev-Sharpened Cosine Filters

^{1}Department of Electronics, INAOE, 72840 Tonantzintla, PUE, Mexico^{2}Cátedras-CONACYT, ESCOM-IPN, 07738 Mexico City, DF, Mexico

Received 27 May 2014; Accepted 12 January 2015

Academic Editor: Changchun Hua

Copyright © 2015 Miriam Guadalupe Cruz Jiménez 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 prove that the Chebyshev sharpening technique, recently introduced in literature, provides filters with a Minimum Phase (MP) characteristic when it is applied to cosine filters. Additionally, we demonstrate that cascaded expanded Chebyshev-Sharpened Cosine Filters (CSCFs) are also MP filters, and we show that they achieve a lower group delay for similar magnitude characteristics in comparison with traditional cascaded expanded cosine filters. The importance of the characteristics of cascaded expanded CSCFs is also elaborated. The developed examples show improvements in the group delay ranged from 23% to 47% at the cost of a slight increase of usage of hardware resources. For an application of a low-delay decimation filter, the proposed scheme exhibits a 24% lower group delay, with 35% less computational complexity (estimated in Additions per Output Sample) and slightly less usage of hardware elements.

#### 1. Introduction

A Minimum Phase (MP) digital filter has all zeros on or inside the unit circle [1]. We consider in this paper MP Finite Impulse Response (FIR) filters, which find applications in cases where a long delay, usually introduced by Linear Phase (LP) FIR filters, is not allowed. Examples of these cases include data communication systems or speech and audio processing systems [2–4].

The basic building block analyzed in this paper, the* cosine filter*, is a simple FIR filter whose transfer function and frequency response are, respectively, given by This filter is of special interest because of the following main reasons: (a)It has MP property because its zero lies on the unit circle.(b)It has a low computational complexity because it does not require multipliers, which are the most costly and power-consuming elements in a digital filter [5].(c)It has a low usage of hardware elements, which can be translated into a low demand of chip area for implementation.

The fact that a cosine filter has the Minimum Phase characteristic becomes significant because these basic building blocks can be used to design filters with a low delay and simultaneously a low computational complexity and a low usage of hardware resources.

Due to the aforementioned characteristics, cascaded expanded cosine filters were investigated in [6]. A* cascaded expanded cosine filter* is defined as a filter with transfer function and frequency response, respectively, given bywith being the cosine filter given in (1), whereas and are arbitrary integers. In method [6], Rouche’s theorem was employed to demonstrate that when cascaded expanded cosine filters are sharpened with a modified version of the technique from [7] (originally devised for LP FIR filters), the result is an overall FIR filter that has all its zeros on or inside the unit circle; that is, it satisfies the MP characteristic. Nevertheless, the resulting filter still has a high group delay because a large number of cascaded expanded cosine filters are needed to meet a given attenuation specification, as we see in the examples from [6].

On the other hand, the Chebyshev sharpening technique was recently introduced in [8] to design LP FIR filters based on comb subfilters for decimation applications. In that method the sharpening is performed with an th degree Chebyshev polynomial of first kind, defined as where are integers [9]. When applied to comb filters, Chebyshev sharpening provides solutions with advantages like a simple and elegant design method, a low-complexity resulting LP FIR filter, and improved attenuation characteristics in the resulting filter. However, filters from [8] are not guaranteed to have MP characteristic.

From the aforementioned literature we can extract the following observations:(a)In MP FIR filters the reduction of the group delay is a priority.(b)The use of cosine filters results in low-complexity multiplierless MP FIR filters.(c)The recent Chebyshev sharpening method from [8] can improve the attenuation of cosine filters and is a potentially useful approach to preserve a simple multiplierless solution with a lower group delay in comparison with simple cascaded expanded cosine filters.

Motivated by the remarks listed above, we present in this paper the following contributions:(1)The mathematical proof that the use of Chebyshev sharpening in cosine filters, which produces* Chebyshev-Sharpened Cosine Filters* (CSCFs), guarantees resulting multiplierless filters with all of their zeros placed on the unit circle, that is, with MP property: this demonstration hinges upon the factorization of the transfer function of the CSCF into second-order sections, taking advantage of the antisymmetry of the roots of the Chebyshev polynomial.(2)The mathematical proof that* cascaded expanded CSCFs* have also all of their zeros placed on the unit circle: this demonstration is a simple extension of aforementioned proof for CSCFs.(3)The explanation of how cascaded expanded CSCFs can be efficiently employed in the design of MP FIR filters.(4)Examples where it is shown that cascaded expanded CSCFs are useful to obtain the same stopband attenuation as cascaded expanded cosine filters but with a lower group delay: from these examples we see an improvement from 23% to 47% in the reduction of the group delay, at the cost of a slight increase of the usage of hardware resources. For an application in a decimation filter embedded into a low-delay oversampled Analog-to-Digital Converter (ADC), the proposed scheme exhibits a 24% lower group delay referred to high rate, with 35% less computational complexity (estimated in Additions per Output Sample) and slightly less usage of hardware elements.

Following this introduction, Section 2 presents the definition of CSCFs and cascaded expanded CSCFs. The proofs of MP characteristic in CSCFs and cascaded expanded CSCFs are given in Sections 3 and 4, respectively. In Section 5 we provide details on the characteristics and applications of the cascaded expanded CSCFs. Then, Section 6 presents examples and discussion of results. Finally, concluding remarks are given in Section 7.

#### 2. Definition of Chebyshev-Sharpened Cosine Filter (CSCF) and Cascaded Expanded CSCF

We define the transfer function and the frequency response of an th order Chebyshev-Sharpened Cosine Filter (CSCF), respectively, aswithwhere are the coefficients of the Chebyshev polynomial of first kind, represented in (5), and is given in (1). To obtain a low-complexity multiplierless implementation, the constant must be expressible as a Sum of Powers of Two (SOPOT). To this end, we set where denotes “the closest value less than or equal to that can be realized with at most adders” and denotes rounding to the closest integer less than or equal to . To provide an improved attenuation around the zero of the cosine filter, must be as close as possible to its upper limit [8]. This is achieved by increasing the integer . The value in (8)-(9) is usually set as an integer equal to or greater than 2 for applications in decimation processes [8]. However, we will allow for more flexibility to the parameter in this paper, as will be explained in the next section.

The transfer function and frequency response of a cascaded expanded CSCF are, respectively, defined aswhere the integer indicates the number of cascaded CSCF blocks, each of them repeated times, with . Every value of is a distinct factor that expands a different CSCF whose corresponding order is . These CSCFs have different factors , which can be obtained using (9), just replacing by and by , where and are integer parameters that correspond to the th CSCF in the cascade. Figure 1(a) shows the structure of the CSCF, where we have that , with and with if is odd or if is even. Dashed blocks in Figure 1(a) appear only if is odd. Figure 1(b) presents the structure of the cascaded expanded CSCF whose transfer function is given in (10).