Design Of Maximally Flat Filters For Signal Processing .
Leena Samantaray, Rutuparna Panda,Sanjay AgrawalWSEAS TRANSACTIONS on SIGNAL PROCESSINGDesign of Maximally Flat Filters for Signal Processing ApplicationsLEENA SAMANTARAY1, RUTUPARNA PANDA2, SANJAY AGRAWAL21Department of Electronics and Communication Engineering,Ajay Binay Institute of Technology, Cuttack, INDIA2Department of Electronics and Telecommunication EngineeringVeer Surendra Sai University of Technology, Burla, INDIAleena sam@rediffmail.com, r ppanda@yahoo.co.in, agrawals 72@yahoo.comAbstract: - Existing window based methods used for the design of finite impulse response filters suffer frompossessing characteristics like the minimum sidelobe energy and the maximally flatness within the passband.The proposed method solves these problems. This paper presents a generalized window based approach to thedesign of maximally flat finite impulse response filters. The basis functions, which very closely approximatesthe prolate spheroidal wave functions, are explored. The novelty of this proposal is the use of these basisfunctions for designing the filter. Further, an additional parameter ‘a’ is incorporated to control the filterspecifications. An explicit formula for computation of the frequency response is derived, which is a newcontribution. It is shown that the impulse response coefficients of the maximally flat filter can be obtaineddirectly from the frequency response. Frequency domain characterization is made. Further contribution is tomeet the given filter specifications with lower order generalized prolate type window functions. Statisticalanalysis is performed to validate the proposed method. The proposed maximally flat filter outperforms thestate-of-the-art methods. Finally, it is concluded that the proposed filter exhibits better error convergence andtracking performance, which may be useful for precision filtering of biomedical signals because of their lowpassband and stopband errors.Key-Words: - Digital filter design, maximally flat filters, prolate spheroidal wave functions, t-Test.implementation of digital filters. During the pastfew decades, different window functions have beendeveloped for the design of FIR filters [5,6]. TheKaiser [7,8] window based FIR filters is useful forapproximating the minimum sidelobe energy in themagnitude response.Dolph–Chebyshev [9]window based design approximates the minimumpeak sidelobe ripple. In this connection, it ispertinent to mention here that B-spline windowfunctions are found to be more widespread, becausethey very closely follow the looked-for frequencyresponse [10-13]. Rectangular, Hanning andHamming window based FIR filter performancesare analyzed in [14]. These window functionsexhibit sidelobes, which is not desirable for highprecision filtering applications. Analysis of theBlackman window and Flat top window based FIRlowpass filter is presented in [15]. The Blackmanwindow based filter has shown higher sidelobe rolloff rate than the Flat top window based design.From the analysis, it is seen that the Flat topwindow based design provides a better filteringeffect compared to the Blackman window basedmethod [15].In [16], the authors have suggested an adjustablewindow for the design of FIR filter. They have1 IntroductionDigital filters play a significant role in signalprocessing. Due to the advent of digital signalprocessing (DSP) techniques, there is a strong needto design and develop efficient digital filters. In thiscontext, a large number of research papers appeareach year with continuous perfection in the digitalfilter design methods. Digital filter is an importantfragment of the digital signal processing system.Filters are classified into two types; i.e. FIR and IIRdepending on the form of filter equations and thestructure of the implementation [1, 2]. Windowbased technique and frequency sampling techniqueis two mostly used methods of FIR filter design.Requirements of minimum ripples in the passbandand the stopband, stopband attenuation [3,4] andtransition width decides a design criterion. In thepast, researchers have proposed numerousprocedures for the design of digital filters. It isnoteworthy to say that FIR filter offers numerousbenefits than IIR filter. Recently, researchers ponderthe design of FIR filters as a demanding yetchallenging problem.The signal processing researchers have showntremendous effort to improve accuracy, sidelobereduction, processing speed and ease inE-ISSN: 2224-348855Volume 15, 2019
Leena Samantaray, Rutuparna Panda,Sanjay AgrawalWSEAS TRANSACTIONS on SIGNAL PROCESSINGstability of a digital filter and, thus, in turn, it isnever feasible to implement a stable FIR filter.Hence, the truncation of the convolution series isrequired. The causal FIR filter obtained by simplytruncating the convolution series shown in Eq. (2)exhibits an oscillatory behavior, which is known asthe Gibb’s phenomenon. The oscillatory nature orthe number of ripples in the filter response increaseswith the increase in the order of the filter.Owing to the above-mentioned reasons, the FIRdigital filter designers concentrate on the alternativeapproaches to obtain the impulse responsecoefficients h [ n ] or H (ω ) that approximates thedesired response. The best alternative is the directdesign of FIR filter coefficients. The FIR filtercoefficients are directly obtained from the frequencyresponse equation displayed in Eq. (1),where H (ω ) is the desired response. The directdesign method of FIR filter coefficients has beenimplemented since 1999 [10]. The improvement inthe response of filters based on the direct designreported in [10] has been found superior to otherfilter design methods. However, results wereobtained based on more or less heuristics. They [10]have chosen fractional order (for example ρ 3.139 )heuristically to get the desired specifications insteadof considering the order of filters to be pure integernumbers. B-spline interpolation techniques arepopular because of certain properties of B-splinepolynomials. Generalized B-spline interpolationschemes are more popular, because of additionalsmoothing. A brief idea about B-spline interpolationis discussed in [11]. Note that the sidelobe energy(of the magnitude response of the filter underconsideration) can be greatly minimized by usingthe generalized B-spline basis functions. Thesidelobe ripples are also significantly reduced byusing these window functions.It is noteworthy to mention here that theinvestigation of a signal, which is both time andband-limited, is very challenging. Solving theunsolved problems is given more significancenowadays. In this connection, exploring the beautyof prolate spheroidal wave functions (PSWF) is aworthwhile subject of study. Prolate meaning is –having a polar diameter of larger length than theequatorial diameter. Note that the PSWF are a set offunctions derived from time-limiting and lowpassing followed by a second time-limitingoperation. The PSWF are the time-limited functions.Slepian, Landau and Pollak [19] performedpioneering work in this area. It is important to claimthe fact that the PSWF play an important role insolving different engineering problems. Thesecombined a fixed window, tan hyperbolic functionand a weighted cosine series. The weighted cosineseries is simply multiplied by a variable. Theyhave compared the performance of their windowwith the hamming and Kaiser window. Authors in[17] presented an adjustable window combiningBlackman and Lanczos windows. The performanceis compared with Gaussian, Lanczos and Kaiserwindows. All these three windows have somecontrol mechanism. Design of a lowpass FIR filterusing Nuttall, Taylor and Tukey windows has beenpresented in [18]. Their behavior with adjustablewindow based design has not been compared.However, the above mentioned methods have moreor less concentrated on the reduction of sideloberoll-off rate. These windows based design methodsexhibit better results when the order of the windowis very high. They have never focused on the designissue of maximally flat filters. They are also silentabout the direct implementation of FIR filters.Nevertheless, recently the direct implementationof the FIR filters has shown much promise forgaining the maximally flat (MF) characteristics.These methods also provide us minimum energy inthe sidelobe of the magnitude response. Thesedirect implementation methods have shown definiteprogress above the window based design methodsfor implementation of the digital FIR filters. Thereason is that the maximum sidelobe ripple of aparticular window is fixed. The stopbandattenuation is also fixed. Usually one encounterstwo critical issues while designing filters – i)control over the passband ripple, ii) control over thestopband ripple. These are a few flaws of thewindow based design methods. Note that for aspecified attenuation specification, it is veryimportant to search for an effective windowfunction. We know that the impulse response (foran FIR) for a given desired H (ω ) is given byh [ n] 12ππ H (ω )ejwndω(1) πFor discrete case the filtered output is given as y [k ] x [ k n] h [ n](2)n The Gibb’s phenomenon of truncating the aboveconvolution series (Eq. (2)) is well known from theliterature. Hence, the ideal filters are notimplementable in practice. Several problems arisewith the design and implementation of ideal filters.It is noteworthy to mention here that the impulseresponse of an ideal filter hd ( n ) decreases as ‘n’increases. This is one of the major problems inimplementing ideal filters in reality. This limits theE-ISSN: 2224-348856Volume 15, 2019
Leena Samantaray, Rutuparna Panda,Sanjay AgrawalWSEAS TRANSACTIONS on SIGNAL PROCESSINGThe desired frequency response can be obtainedby using the following relation:PSWF also play significant role in designing moreideal lowpass filters for signal processingapplications. This is the motivation behind thisproposal. In this work, we explore a set of basisfunctions, which very closely approximates thePSWF (Ref. Appendix A).This has motivated us to design MF filters usingspheroidal type generalized B-spline basis functions.Further motivation is the direct design. This paperpresents a more generalized approach to designmaximally flat filters with an additional parameterwhich is an improvement over the earlier methods[10,14-18]. The additional parameter is useful tocontrol the filter response. The frequency responseof the proposed MF filters has been compared withstate-of-the-art methods. From the results given inthe paper, it is observed that the suggested MF filtershows better error convergence and MFcharacteristics. It is believed that the proposedfilters are very useful for signal processingapplications.The organization of the paper is as follows: Abrief introduction to direct design of FIR filters hasbeen given in the introduction section. Section 2deals with the proposed method. Section 3 presentsthe results and discussions. The paper concludes inSection 4.H (ω ) H I (ω ) φ (ω ) φ (ω ) φ (ω )(3) H I (ω ) ψ (ω )where * denotes the convolution, H I (ω ) is the ideallow pass filter response and ψ (ω ) is the requiredconvolving function. Note that α and ρ are thecontrolling parameters. As per the above theoreticaldiscussions, it is clear that ρ is an integer anddepends on the number of poles of the transferfunction of the filter. This impose a limitation onfurther improvement of the filter response. This hasmotivated the author to develop a more generalizedapproach to develop digital filters. The use of twosided Laplace transform is very common in signalprocessing. The main objective of this work is tointroduce another controlling parameter ‘a’ in thedesign process. With a proper choice of the tuningparameter ‘a’, the desired response can be obtainedwithout disturbing the order (ρ) of the filter to anearby fractional number.In this section, we propose a generalized windowfunction by compelling an exponentially decreasingfunction instead of a rectangular pulse as displayedin Fig.1.2 Proposed MethodThe proposed method is an attempt to approximatethe rectangular shape frequency response of an ideallow pass filter. It is noteworthy to point out the facthere that the direct method smooths out the sharptransition curve. Direct design approaches toimplement FIR filters are becoming popular and usetrapezoidal filters. In this section, we propose adirect approach to design a digital FIR filter using ageneralized B-spline window function [12]. In thisapproach, a second order (ρ 2) B-spline functionhas been proposed. The window function isconstructed by convolving a pulse (rectangularshape) of width 2αωc / ρ and height π / αωc withitself. The focus is to obtain a smooth curve in thetransition band. Additional smoothing in thetransition band is achieved by repeatedly convolvingtwo rectangular shaped window functions (width ofeach pulse is half and the height of each pulse istwice with respect to the original one). Thisapproach is repeated and the ρth order windowfunction is generated. The newly proposed windowfunction is then convolved with the ideal filterfrequency response (rectangular shape) to substitutethe sharp transition curves.E-ISSN: 2224-3488Fig.1. The window function.Here, a group of generalized window functions isconstructed following the basic idea that thegenerating function used to construct the windowfunction of degree ‘n’ has (n-1) continuousderivatives. Note that these functions also satisfy alinear differential equation of order (n 1). Ageneralized window function can be defined as [12].The polynomial B n1 ( x) is a particular case of thesegeneralized windows. The following relation ofB nr ( x) is obtained.n 1nrB ( x) N w j g nr ( x x j ) u ( x x j )j 0(4)where n is the degree, r is the type, N is the constantof normalization, and u(x) refers to a unit stepfunction. Note that g nr ( x) is the creating function, wjare coefficients of multiplication in the jth slice ofthe window function. The number of generalized57Volume 15, 2019
Leena Samantaray, Rutuparna Panda,Sanjay AgrawalWSEAS TRANSACTIONS on SIGNAL PROCESSINGsymmetry. The conditions are s 2 0 , s a ,s ja . In this filter development, we consider thethird case, i.e. s ja .Similarly, a large number of windows are1constructed.A particular window is chosen from awk n 1(6)set of functions for designing the MF filter. The jj 0k ( x x j )frequency response closely approximates the desiredg(x) xn(7)response by increasing the degree of the windowThe generalized window considered here is real.functions. In this method of filter design, theThe generating function considered for the filterproposed window is used as the convolving functiondesign is a solution of a differential equationφ (ω ) of Eq. (3). The convolving function for thesatisfying continuity up to ( ρ 2)th order. Thewindow is expressed asgenerating function also satisfies required initial ρ ω π ρconditions. Note that initial conditions in twosided 1 k B(ω ) N ρwk g (10) αωαω2Laplace transform play important role in analysis ofc k 0 c system functions. The poles of the generalized BB(ω ) is a polynomial well-defined between thespline window ( B(ω ) ) must be symmetrical aboutinterval [- αωc to αωc ]. For other interval values, itthe real axis. To make the proposed windowis zero. Note that H (ω ) (Eq. (3)) is obtained bysymmetrical about the center, the ensuing mirrorsymmetry condition is enacted.repeated convolution. The function used is displayedg ( x ) g ( x)a(8)in Fig. 2. Here, N . The corresponding(1 e a )In order to satisfy Eq. (8), ( B(ω ) ) is symmetricalωk values are displayed in [12].function. It possesses symmetry about the imaginaryaxis. Thus, for the proposed window, the poles of( B(ω ) ) possesses four quadrant symmetry.The proposed window function of any order canbe calculated from a zero-degree window shown inFig.1. Using the convolution properties discussed in[12], various types of windows are generated. Theprincipled feature of the proposed functions is thatFig. 2. The convolving function.these functions give compact support. One cangenerate many such types of windows for a degreeThe filter response (lowpass) achieved by smearingmore than 3. The generating function, which isthe process H I (ω ) B(ω ) is written asconsidered as the seed, is g 01 ( x) e( a x ) , where ‘a’ isthe scale factor. The proposed idea is different from ρ ρ ω ωcN ρ 1 k (11)H (ω ) 1 ( 1) k wk the design proposed in [10], where the generatingρ ! k 0 k 2 αωc function is a rectangular function. Whereas in thiswork, an exponentially decreasing window is usedThe discrete time impulse response is achieved byto generate the first generating function in the series.evaluating the inverse Fourier transform of Eq. (11).An additional parameter ‘a’ called scale factor isThe discrete time impulse response of the lowpasshosted. Interestingly, this factor guides the user forfilter isapproximating the filter bandwidth competently.h [ n] The proposed window of zero degree iswindow functions are more for a degree n 3 [12].For a given degree n, many types r of generalizedwindows can be generated.N n 1(5)a e ax u ( x) e a e a ( x 1) u ( x 1) B ( x)(1 e a ) 01ωc sin(nωc ) a 2 ( cos(n / ( M 1)) cos a )(12)ρ( sinc ) π (nωc ) 1 cos a a 2 ( n / ( M 1) )2(9) n where sinc sin( pi x) ( pi x) and x M 1 The magnitude persists unity value all throughthe passband and zero all through the stopbandfrequencies. Hence, these filters are calledmaximally flat filters. We use an ideal filter, whichis applied before B(ω ) . This is vital to escape theIt is interesting enough to note here that whena 0 , this approaches to a zero order windowproposed in [12]. This is same as that of a centeredrectangular pulse. The first order generalized Bspline window can be generated by recognizing thefact that the poles of ( B(ω ) ) have four quadrantE-ISSN: 2224-348858Volume 15, 2019
Leena Samantaray, Rutuparna Panda,Sanjay AgrawalWSEAS TRANSACTIONS on SIGNAL PROCESSINGparameters. The number of convolution operationswere controlled using the filter order ‘ρ’. As a result,the shape of the filter response is controlled.In this context, a new method for the design ofMF filter with improvement mechanism is explored.The key to the improvements claimed is theinclusion of an additional control parameter ‘a’. Theerror in the stop band is minimized drastically whilepreserving the other specifications of the filter underconsideration. The control parameter ‘a’ provesbetter to control the flatness of the desired responseof the filter. This proposal has certain advantagesover other methods.sharp transition. In this work, a lowpass filter isdesigned. Similarly, the discrete time impulseresponse for an ideal highpass, bandpass, andbandstop filter can also be designed. This kind ofdirect approach may be useful for many meansquare signal-processing applications.In this paper, standard definitions for cut offfrequency ωc , stopband attenuation As, and transitionwidth ω are considered [6].Note that the stopbandattenuation As 20 log10 (δ s ) dB and the cutofffrequency ω (ωs ω p ) / 2 .cIt is important to design MF filters for meansquare signal processing applications. A real-worldmethodology is to multiply the ideal impulseresponse hd (n) by an appropriate window w(n) withfinite duration. As a result, the decay is faster. Theimpulse response reduces to a zero value quickly.The middle part of the impulse response can be usedfor designing a linear phase FIR filter. For anincreased filter length, the ripples never vanishwhen we use a rectangular window. However, anon-rectangular window lessens its magnitudesmoothly. One can see many single parameterwindow functions in the literature [14-18].Examples include – Hanning, Hamming, Blackman,etc. These window functions have the control overthe transition width ω of the f
Digital filters play a significant role in signal processing. Due to the advent of digital signal processing (DSP) techniques, there is a strong need to design and develop efficient digital filters. In this context, a large number of research papers appear each year with continuous perfection in the digital filter design methods.
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Designing FIR Filters with Frequency Selection Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Differentiation Designing IIR Filters with Impulse Invariance Designing IIR Filters with the Bilinear Transform Related Analog Filters. Lecture 22: Design of FIR / IIR Filters. Foundations of Digital .
all-digital receiver systems, the Shannon sampling scheme is usually implemented by using one ADC. In this case, the wideband FD interpolation lter using the . Maximally Flat and Least-Square Co-Design of Variable Fractional Delay Filters for Wideband Software-De ned-Radio 5 The ideal lter response in Eq. (1) can be approximated using the .
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IIR filters are digital filters with infinite impulse response. Unlike FIR filters, they have the feedback (a recursive part of a filter) and are known as recursive digital filters. Figure 2 Block diagrams of FIR and IIR filters For this reason IIR filters have much better frequency response than FIR filters of the same order. Unlike FIR .
domain to obtain a digital filter that meets the specifications. The commonly used analog filters are 1. Butterworth filters - no ripples at all, 2. Chebychev filters - ripples in the passband OR in the stopband, and 3. Elliptical filters - ripples in BOTH the pass and stop bands. The design of these filters are well documented in the literature.
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