Design of Cut Off-Frequency Fixing Filters by Error Compensation of MAXFLAT FIR Filters

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Article Design of Cut Off-Frequency Fixing Filters by Error Compensation of MAXFLAT FIR Filters

Daewon Chung , Woon Cho, Inyeob Jeong and Joonhyeon Jeon *

Division of Electronics & Electrical Engineering, Dongguk University-Seoul, Seoul 04620, Korea; [email protected] (D.C.); [email protected] (W.C.); [email protected] (I.J.) * Correspondence: [email protected]; Tel.: +82-2-2260-3545; Fax: +82-2-2285-3343

Abstract: Maximally-flat (MAXFLAT) finite impulse response (FIR) filters often face a problem of the cutoff-frequency error due to approximation of the desired frequency response by some closed-form solution. So far, there have been plenty of efforts to design such a filter with an arbitrarily specified cut off-frequency, but this filter type requires extensive computation and is not MAXFLAT anymore. Thus, a computationally efficient and effective design is needed for highly accurate filters with desired frequency characteristics. This paper describes a new method for designing cutoff-frequency-fixing FIR filters through the cutoff-frequency error compensation of MAXFLAT FIR filters. The proposed method provides a closed-form Chebyshev polynomial containing a cutoff-error compensation function, which can characterize the “cutoff-error-free” filters in terms of the degree of flatness for a given order of filter and cut off-frequency. This method also allows a computationally efficient and accurate formula to directly determine the degree of flatness, so that this filter type has a flat magnitude characteristic both in the passband and the stopband. The remarkable effectiveness of the proposed method in design efficiency and accuracy is clearly demonstrated through various examples, indicating that the cutoff-fixing filters exhibit amplitude distortion error of less than 10−14 Citation: Chung, D.; Cho, W.; Jeong, and no cut off-frequency error. This new approach is shown to provide significant advantages over I.; Jeon, J. Design of Cut the previous works in design flexibility and accuracy. Off-Frequency Fixing Filters by Error Compensation of MAXFLAT FIR Keywords: maximally flat filters; cutoff-frequency error; cutoff-frequency error compensation; linear Filters. Electronics 2021, 10, 553. phase filters; digital filters; cut off-frequency fixing filters https://doi.org/10.3390/electronics 10050553

Academic Editor: Flavio Canavero 1. Introduction Received: 21 January 2021 The theory, design, and application of a finite impulse response (FIR) filter with math- Accepted: 24 February 2021 ematical analysis have been widely studied [1–23]. Among them, especially, maximally-flat Published: 26 February 2021 (MAXFLAT) FIR filters, which are known for their design simplicity, accuracy, and high stopband attenuation, have an important role in some applications, such as waveform trans- Publisher’s Note: MDPI stays neutral mission and audio/image processing [3–14]. The basic idea for the design of MAXFLAT with regard to jurisdictional claims in FIR filters is based on the well-known closed-form transform function and is realized published maps and institutional affil- due to its closed-form polynomial [7–9,15–18]. However, previous MAXFLAT FIR filter iations. designs involve approximation of the desired frequency response by some closed-form polynomial, like a Hermite Interpolation [15], Miller’s conformal mapping [16], Bernstein polynomial [10,17], Fahmy’s Integral method [18], or Krawtchouk polynomial [19], which is then mapped to the filter function by certain transformations. The MAXFLAT FIR filter Copyright: © 2021 by the authors. by any closed-form polynomial does not have any independent (“free”) parameters. This Licensee MDPI, Basel, Switzerland. is due to the fact that the maximum possible number of zeros at z = ±1 is imposed, This article is an open access article which leaves no degree of freedom, and, thus, no independent parameters. Thus, there distributed under the terms and is no direct control over the frequency response to obtain such a filter with a prescribed conditions of the Creative Commons specified cut off-frequency. For that reason, the magnitude response of this filter type Attribution (CC BY) license (https:// never passes through the desired cut off-frequency [19–22]. In order to overcome such creativecommons.org/licenses/by/ a difficulty, there have been many efforts for comprising independent parameters in a 4.0/).

Electronics 2021, 10, 553. https://doi.org/10.3390/electronics10050553 https://www.mdpi.com/journal/electronics Electronics 2021, 10, 553 2 of 15

closed-form function and, then, frequency response by (modified) closed-form function is controllable by these parameters. FIR filters using such a closed-form function including controllable parameters are no longer MAXFLAT, and, thus, various design methods have been published recently to obtain a best frequency response [7–14,23]. Jeon et al. have reported a closed-form least error gain method [23] for the design of FIR filters with both a flat magnitude and exact cut off-frequency. However, this method has the disadvantage of requiring a lot of computation processing for estimating the order of flatness, so that the error frequency function is closest to a zero gain at all frequencies [23]. It also sometimes has the frequency response distortion by wrongly determining the order of flatness. Huang et al. have proposed a closed-form weighted least square design to obtain FIR filters with exact cut off-frequency. However, this method has the disadvantage of providing non-negligible amplitude distortions of more than 4.2% by the Gibbs effect despite allowing a large amount of computation for estimating a convolution window [7,8]. Tseng and Lee introduced a closed-form design by mapping three discrete transforms, but there is a serious problem, such that filter cutoff-frequency error rapidly increases with growing cut off-frequency [9]. Recently, Roy et al. have reported Chebyshev closed-form FIR filters using a Bernstein polynomial [10]. They can reduce the filter design complexity by using a specific threshold value, but non-negligible amplitude distortions are in the stopband and passband. Hence, a computationally efficient design is needed for highly accurate filters with desired frequency characteristics, i.e., magnitude response exactly passes through prescribed, specified cutoff-frequency and is maximally flat in both the passband and the stopband. The objective of this paper is to present a new method for the design of cut off- frequency fixing filters using a “cut off-error-free” polynomial function. For this pur- pose, a frequency-response error compensation function between the desired and actual frequency responses is derived through the generalization of the three closed-form polynomials [10,15–17] into a Chebyshev polynomial form. The proposed method provides a closed-form Chebyshev polynomial to characterize this filter type in terms of the degree of flatness for a given order of filter and cut off-frequency. Then, the magnitude response passes exactly through the prescribed specified cut off-frequency, and the filter also has independent (“free”) parameters that permit direct control over the frequency response, i.e., there is a tradeoff between the transition bandwidth and the magnitude flatness [24]. Finally, to determine the degree of flatness, so that a (cut off-frequency fixing) filter has a flat magnitude response characteristic for a given order of filter and cut off-frequency, a computationally efficient and accurate formula is derived from the cut off error compensation function. This paper is organized as follows. In Section2, we propose a closed-form error function for the cut off-error compensation of MAXFLAT FIR filters and introduce a closed- form Chebyshev polynomial to design cut off-frequency fixing filters. In Section3, through the analysis of this error compensation function, we provide an exact and direct expression to choose a cutoff-frequency fixing FIR filter with a desired frequency response. Design examples that demonstrate the power of the new technique are shown. Conclusions are drawn in Section4.

2. Closed-Form Chebyshev Polynomial for Cutoff-Frequency Fixing Filter Design The transfer function H(z) of a symmetric linear-phase FIR ﬁlter with the impulse response hn of order 2N is written as:

2N ( N ) −n −N H(z) = ∑ hnz = z hN + ∑ hN−nTn(ω) (1) n=0 n=1 Electronics 2021, 10, 553 3 of 15

where Tn(ω) is the Chebyshev polynomial of the ﬁrst kind of degree n, and the independent transformed variable w [25] is related to the digital domain by:

1 −1 w = z + z jω = cosω (2) 2 z=e

confined to the interval [−1, 1]. The frequency response Hejω of the filter can be expressed as: H ejω = e−jNωQ(cosω) = z−N Q(w) (3) z=ejω by using the zero-phase transfer function Q(w), which represents a polynomial of the real variable w = cosω, (0 ≤ ω ≤ π). Then, the filter can be said to be lowpass and MAXFALT if Q(cosω) has the following properties:

Q(cosω)|ω=0 = 1 (4) ∂vQ(cosω) = = ( − ) + v 0, v 1, 2, . . . , 2 N K 1 (5) ∂ω ω=0 ∂vQ(cosω) = = − v 0, v 0, 1, 2, . . . , 2K 1 (6) ∂ω ω=π where N − K and K represent orders of ﬂatness at ω = 0 and ω = π, respectively. It is shown, based on the results of [10,15–17], that such a lowpass function can be accom- plished by using one of the closed-form polynomials, namely, Hermite Interpolation polynomial [15], Miller’s conformal mapping function [16], and Bernstein polynomial [10,17], which are expressed, respectively, as:

N−K K + i − 1 P (x) = (1 − x)K xi, 0 ≤ x ≤ 1 (7) N,K ∑ i i=0

1 N ( ) = N−K 2i ≤ 2 ≤ HN, K jΩ N ∑ i=0 Ω , 0 Ω 1 (8) (1 + Ω2) i N − B ( f ; x) = N−K xi(1 − x)N i, 0 ≤ x ≤ 1 (9) N,K ∑ i=0 i A where = A!/((A − B)!B!) and cosω = 1 − 2x = 1 − Ω2 / 1 + Ω2 . The work B in [10,17] reported a functional equivalence between Equations (7)–(9). Kaiser [26] also established a link between Herrmann’s polynomial Equation (7) and Fahmy’s Integral [18]. Consequently, this implies that all MAXFLAT FIR ﬁlters published so far can be expressed as one and the same closed-form solution. By using the transformations x = (1 − w)/2 and Ω2 = (1 − w)/(1 + w) on Equations (7)–(9), such a ﬁlter can be expressed by a generalized closed-form function Qg(w) as below.

Qg(w) = PN,K(x) 1−w = Ha(jΩ) 2 1−w = BN,K( f : x) x= 2 Ω = 1+w 1−w x= 2 K N−K i (10) 1+w K + i − 1 1−w = 2 ∑ 2 i=0 i

The Chebyshev polynomial form of Equation (10) can be obtained as: ( ) 1 + w K N−K Qg(w) = gN−K + 2 ∑ gN−K−iTi(w) (11) 2 i=1 Electronics 2021, 10, x FOR PEER REVIEW 4 of 15

Electronics 2021, 10, 553 4 of 15

푖 푖 1 − 푤 1 2푖 2푖 by using the Chebyshev( ) representation= {( ) + equation:2 ∑(−1)푢 ( ) 푇 (푤)} (12) 2 22푖 푖 푖 − 푢 푢 ( 푢=1 ) 1 − w i 1 2i i 2i = + 2 (−1)u T (w) (12) 푔2i i ∑ i − u u where interpolation coefficients2 2 푖 are expressedu =as1 [23]: 푖 where interpolation coefficients푁−퐾−푖 gi are1 expressed푁 − asℓ − [231]: 2(푁 − 퐾) − 2ℓ 푔푖 = (−1) ∑ ( ) ⋅ ( ) (13) 22(푁−퐾−ℓ) 퐾 − 1 푖 − ℓ i N−K−i ℓ=0 1 N − l − 1 2(N − K) − 2l gi = (−1) · (13) ∑ 22(N−K−`) K − 1 i − l Computation of ℎ푛′푠 in Èquation=0 (1), using Equation (12), is reported in [23,27,28]. In a similar way, by mapping0 Equation (11) to Equation (1), a relational formula between ′ Computation′ of hn s in Equation (1), using Equation (12), is reported in [23,27,28]. In ℎ푛 푠 and 푔푖 푠 can be obtained as: 0 a similar way, by mapping Equation (11) to Equation (1), a relational formula between hn s 0 푁−퐾 푁−퐾 and gi s can be obtained1 as: 2퐾 2퐾 ℎ푛 = 2퐾 { ∑ ( ) 푔푖 + ∑ ( ) 푔푁−퐾−푖} (14) 2 (N−K푛 − 푖 N−K푛 − 푖 − (푁 − 퐾) ) 1 푖=0 2K 푖=1 2K hn = g + g − − (14) 퐴 22K ∑ n − i i ∑ n − i − (N − K) N K i where ( ) = 0 if 퐴 < 퐵 ori= 0퐵 < 0. It can be seeni=1 from Equation (11) that, for a given 푁, 퐵 there are 푁 possibleA filters (with 푁 different cutoff points) corresponding to 퐾 = 1 to where = 0 if A < B or B < 0. It can be seen from Equation (11) that, for a 푁. Thus, to determineB 퐾 suitable to a desired cutoff point of 푤푐(−1 < 푤푐 < 1), existing methodsgiven [N15,– there18,20] aretypicallyN possible use the filters empirical (with expressionN different (Herrmann cutoff points) used this corresponding term with- to outK explicitly= 1 to N defining. Thus, to the determine relation (K[15suitable], Equation to a desired(11))) given cutoff by point Herrmann of wc( −[151 ]< asw fol-c < 1), lowsexisting: methods [15–18,20] typically use the empirical expression (Herrmann used this term without explicitly defining the relation1 ([15], Equation (11))) given by Herrmann [15] as follows: 퐾 = 푁 − [ 푁(1 − 푤푐) + 0.5] (15) 2 1 K = N − N(1 − w ) + 0.5 (15) 1 −1 c where 푤 = (푧 + 푧)| 푗휔푐 = 푐표푠휔 for a2 give cutoff-frequency 휔 , and [푝] denotes 푐 2 푧=푒 푐 푐

the integer part 1of 푝−.1 For this reason, the classical designs naturally involve an approxi- where wc = z + z jωc = cosωc for a give cutoff-frequency ωc, and [p] denotes the mate cut off-frequency2 error z =[20,22,23e ]. For example, in the case of 푤 = 0.4 (correspond- integer part of p. For this reason, the classical designs naturally involve푐 an approximate ing to 휔 = 0.369휋 in Reference [15]), Figure 1 shows the MAXFLAT FIR filters, which cut off-frequency푐 error [20,22,23]. For example, in the case of w = 0.4 (corresponding to are chosen according to Equation (10) when using Equation (15).c It is shown that the ω = 0.369π in Reference [15]), Figure1 shows the MAXFLAT FIR filters, which are chosen MAXFLATc FIR filter has both a passband and stopband that are maximally flat, but its according to Equation (10) when using Equation (15). It is shown that the MAXFLAT magnitude response never passes through the desired cut off-frequency. Particularly, as FIR filter has both a passband and stopband that are maximally flat, but its magnitude the order of filter becomes lower, the cut off-frequency error becomes larger. response never passes through the desired cut off-frequency. Particularly, as the order of filter becomes lower, the cut off-frequency error becomes larger.

(45, 32) 1.0

0.8

1 2 (=-3dB) 0.6 (28, 20)



 (20, 14)

g 0.4 wc

Q   (14, 10) c (10, 7) 0.2 (5, 4)

0.0

1.0 0.5 0.0 -0.5 -1.0 w

0.0 0.5 1.0 

Figure 1. Maximally-flat (MAXFLAT) finite impulse response (FIR) filters: Qg(w) with various N for Figure 1. Maximally-flat (MAXFLAT) finite impulse response (FIR) filters: 푄푔(푤) with various wc = 0.4. 푁 for 푤푐 = 0.4.

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One of the main difficulties for MAXFLAT FIR filter design is to express the objective error function in a closed form [27]. This is due to the fact that the MAXFLAT FIR filter by any closed-form polynomial does not have any design (“free”) parameters. Thus, the objective error function has to be derived between the desired and actual variable frequency responses [23,29,30]. Referring to an objective function used in Reference [23], we can define a closed-form error compensation function E(w) between a cut off-frequency fixing filter Q(w) with a desired cut off point at w = wc and the closed-form polynomial filter Qg(w) of Equation (10).

− 1 + w K 1 − w N K E(w) = Q(w) − Q (w) = 22(N−K)C (16) g K 2 2

2(N−K) where 2 is for normalization and CK is a cut off-error compensation factor. Then, CK can be obtained in terms of K and wc as:

1 √ − Qg(wc) C = 2 (17) K K N−K 2(N−K) 1+wc 1−wc 2 2 2 √ by substituting Q(wc) = 1/ 2 into the relation of Equation (16) with w = wc. Conse- quently, adding E(w) into Qg(w) can yield a “cutoff-error-free” filter Q(w) with a cut off-frequency controllable by CK: ( − ) 1 + w K N−K K + i − 1 1 − w i 1 − w N K Q(w) = Q (w) + E(w) = + 22(N−K)C (18) g ∑ i K 2 i=0 2 2 where Equation (18) consists of adding Equation (16) to Equation (10). More importantly, Q(w) passes exactly through the desired√ cutoff point w = wc, that is, Q(w) always satisfies the cutoff condition, Q(wc) = 1/ 2, regardless of the values of K and N. From Equation (18), it can be seen that E(w) is added as an extra term to the (N − K)- th coefficient in the sum (of the second term of Qg(w) given in Equation (10)) without 0 increasing the order of the filter. Then, small changes δgi s of the interpolation coefficients 0 gi s in Equation (13) due to adding E(w) can be obtained as: − − 2(N − K) δg = (−1)N K iC , i = 0, 1, 2, . . . , N − K (19) i K i

0 by transforming Equation (16) into Equation (11). Hence, new gî s caused by adding E(w) to Qg(w) consist of adding Equation (19) to Equation (13). ( i ) − − 1 N − l − 1 2(N − K) − 2l 2(N − K) gˆ = (−1)N K i + C (20) i ∑ 2(N−K−`) K − 1 i − l K i `=0 2 As shown in Equation (18), using E(w) for the cut off-frequency error compensation of Qg(w) leads to a closed-form Chebyshev polynomial, which can characterize cut off-frequency fixing FIR filters as a “cut off-error free” in terms of K. However, the frequency response (i.e., Q(cosω)) of Equation (18) including Equation (16) no longer satisfies Equation (5). This is due to the fact that, for a given N and wc, there are N different “cut off-error free” filters with N different magnitudes responses caused by N different E(w) corresponding to K = 1 to N. For this reason, if a wrong E(w) (i.e., wrong K) is chosen, it will have a negative effect on the flatness of Q(w) even though Qg(w) is MAXFLAT. Hence, it is necessary to select K exactly so that E(w) compensates for the cutoff error of Qg(w) without damaging the flatness in the passband or the stopband. Electronics 2021, 10, 553 6 of 15

3. The Order of Flatness (K) In this section, E(w) given in Equation (18) is analysed for building a computationally efficient and accurate solution to determine an optimal value of K for a given N and wc. As mentioned earlier, for the cut off-frequency error compensation of Qg(w) using E(w) potentially leads to Q(w) with its magnitude response exactly passing through the prescribed cutoff point w = wc. However, the wrong choice of K results in significant magnitude-distortions due to the undesired amplitude of the E(w). Thus, to determine K exactly, so that E(w) compensates for the cut off-frequency error of Qg(w) without damaging the flatness, which is a simple and complete formula-based solution that has to be built on E(w). Actually, in the case of Reference [23] using a distortion error function, they had proposed an iteration training method (see Figure 4 in Reference [23]) of estimating a minimum-distortion error function to determine K for a given N and wc. However, this method has the disadvantage of requiring complicated and enormous computation processing for obtaining a suitable one among N different distortion error functions. This computation complexity increases rapidly with increasing N, which is the order of the filter. In addition, it sometimes has the frequency-response distortion by wrongly determining K. For this reason, a computationally efficient and accurate formula has to be needed for the design of such a filter. From Equation (18), it is seen that E(w) exhibits a “bell-curve-like” shape, which only has one peak gain, and zero gain at w = ±1 (corresponding to ω = 0 and ω = π). Thus, the peak and inflection points of E(w) can play important roles in identifying influential observations in terms of K for a given N and wc. These points can be used to thoroughly characterize the effect of E(w) on Q(w) in terms of K. If wp is the peak point of E(w), wp is given as: 2K − N w = , −1 < w < 1 (21) P N K

from ∂E(w)/∂w|w=wP = 0 and, then, the peak value E(wP) can be obtained in terms of K as: K K K N−K E(w ) = C 1 − (22) P K N N + − by substituting Equation (21) into Equation (16) with wp. In addition, letting w and w I I 2 2 be two inﬂection points of E(w), from ∂ E(w)/∂ w ± = 0, we can get: w=wI p −(2N − 1)(N − 2K) ± N2 − 4(1 − 4N)NK + 4(1 − 4N)K2 w± = , −1 < w± < 1 (23) I 2N2 I ± + − ± where wI (double signs in the same order) denotes wI and wI , and E wI can also ± be obtained in terms of K by substituting Equation (23) into Equation (16) with wI . By ± substituting Equation (21) into Equation (23), wI can be rewritten in terms of wp as:

1 1 q w± = 1 − w ± w2 − 8Kw + 8K, −1 < w± < 1 (24) I 2N P 2N P P I

+ − From Equation (24), it is seen that wI and wI are related to wp by:

w + + w − 1 I I = 1 − w (25) 2 2N P

− + and satisfy the inequality wI < wP < wI . This means that the shape of E(w) is not perfectly symmetric about the line w = wP and can lead to inequalities not only between − + wc and inﬂection points (i.e., wI , wI , and wp) in terms of K but also between E(wc) and − + inﬂection gains (i.e., E wI , E wI , and E wp ). For example, in the case of wc = 0.4 and 0 N = 11, Figure2 shows E(w) s corresponding to K = 3 to 10 and the related parameters are ± indicated in Table1, where CK, wP, and wI have been obtained according to Equations (17), (21) and (23). Figure3 also shows Qg(w), E(w), and Q(w) for each case of the four ranges Electronics 2021, 10, x FOR PEER REVIEW 7 of 15

and 푁 = 11, Figure 2 shows Ε(푤)′푠 corresponding to 퐾 = 3 푡표 10 and the related pa- ± rameters are indicated in Table 1, where 퐶퐾, 푤푃, and 푤퐼 have been obtained according to Equations (17), (21) and (23). Figure 3 also shows 푄푔(푤), Ε(푤), and 푄(푤) for each case of the four ranges described in Table 1 where the prescribed cut off point 푤푐 falls into one − − of the following four ranges due to the choice of 퐾: 푤푐 < 푤퐼 , 푤퐼 < 푤푐 < 푤푃, 푤푃 < 푤푐 < + + 푤퐼 , and 푤퐼 < 푤푐. From these examples, it can be found that the undesired magnitude of Ε(푤) around 푤푝 causes not only overshoot in the passband of 푄(푤) if 푤푐 < 푤푃 (i.e., − − + 푤푐 < 푤퐼 or 푤퐼 < 푤푐 < 푤푃) for a given 퐾, but also undershoot in the stopband if 푤퐼 < 푤푐. However, in these cases, 푄(푤) can have a relatively narrow transition bandwidth due to 퐾. More importantly, it is shown that 푄(푤) has a flat magnitude characteristic in both

the passband and the stopband when Ε(푤푃) in terms of 퐾 is negatively minimized, and, + then, 푤푐 falls in between 푤푃 and 푤퐼 : + 푤푃 < 푤푐 < 푤퐼 (26) This corresponds to 퐾 = 7 in Table 1 and, as shown in Figure 3b, no overshoot and no undershoot appear in the passband and the stopband. For the further evaluation of this result, in the case of 푤푐 = 0.7071 (corresponding to 휔푐 = 0.25휋) and 푁 = 28, Figure 4a, b show Ε(푤) in terms of 퐾 and the effect on 푄(푤), respectively. The related parameters Electronics 2021, 10, 553 7 of 15 are also indicated in Table 2 where 훿푂 and 훿푈 are the peak values of overshoot and undershoot in the passband and the stopband, respectively. It can be seen that taking 퐾 = 23 yields the smallest negative value Ε(푤푃)|퐾=23 (at 푤푃|퐾=23 = 0.6429) and results in + 푤푃|퐾described=23 and 푤 in퐼 | Table퐾=23 1satisfying where the Equation prescribed (26 cut) for off a point given w푤c 푐falls= 0. into7071 one. Consequently, of the following 퐾 = 23 훦(푤)| − − + usingfour ranges potentially due to the leads choice to of K: w퐾c=23< yieldingwI , wI a< cutwc off< -wfrequencyP, wP < fixingwc < w filterI , and + 훿 = 훿 = 0) withw zeroI < overshootwc. From and these zero examples, undershoot it can (i.e., be found푈 that푂 the. Especially, undesired it magnitude is shown that of E (w) 퐾 = 22 and 24 − the filtersaround byw p causes not only have overshoot a tolerant in themagnitude passband distortion of Q(w),if butwc relatively< wP (i.e., narrowerwc < wI or − + transitionwI < bandwc < thanwP) for the a filter given byK ,퐾 but= also23. undershoot in the stopband if wI < wc. However, in these cases, Q(w) can have a relatively narrow transition bandwidth due to K.

3 : point wP x :  point 2 K=10 wI o :  point o wI x 1 ) K=9 K=7

w ( o K=6

 x x o 0 x o x o K=8 x o -1 K=5

-2 K=4 o x K=3  -3 wc

c -4 w 1.0 0.5 c 0.0 -0.5 -1.0  0.0 0.5 1.0 Figure 2. E(w) due to increasing K for N = 11 and wc = 0.4 (ωc = 0.369π). Figure 2. Ε(푤) due to increasing 퐾 for 푁 = 11 and 푤푐 = 0.4 (휔푐 = 0.369휋).

TableTable 1. Related 1. Related parameters parameters of Figure of Figuress 2 and2 and 3. 3.

푲 K Parameters Parameters 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10 CK −0.1982 −0.3355 −0.5405 −0.7333 −0.4836 1.3807 6.7866 17.5271 − 푤퐼 CK −0.7032−0.1982 −0.5507−0.3355 −0.3871−0.5405 −0.2135−0.7333 −0.0300−0.4836 0.1645 1.3807 0.3731 6.7866 0.6037 17.5271 − 푤푃 wI −0.4545−0.7032 −0.2727−0.5507 −0.0909−0.3871 0.0909−0.2135 0.2727−0.0300 0.4545 0.1645 0.6364 0.3731 0.8182 0.6037 + 푤퐼 wP −0.1645−0.4545 0.0300− 0.2727 0.2135− 0.09090.3871 0.0909 0.5507 0.2727 0.7032 0.4545 0.8418 0.6364 0.9583 0.8182 훦(푤 ) + 푐w I −0.2923−0.1645 −0.2886 0.0300 −0.2713 0.2135 −0.2147 0.3871 −0.0826 0.5507 0.1375 0.7032 0.3944 0.8418 0.5941 0.9583 − 훦(푤E퐼 ()w c) −11.7403−0.2923 −2.3584−0.2886 −0.6661−0.2713 −0.2284−0.2147 −0.0550−0.0826 0.0851 0.1375 0.3615 0.3944 1.5264 0.5941 훦(푤 ) − −20.6230 −4.0613 −1.1314 −0.3837 −0.0915 0.1403 0.5898 2.4573 E푃 wI −11.7403 −2.3584 −0.6661 −0.2284 −0.0550 0.0851 0.3615 1.5264 + 훦(푤E퐼 ()w P) −12.5099−20.6230 −2.4404−4.0613 −0.6733−1.1314 −0.2259−0.3837 −0.0531−0.0915 0.0799 0.1403 0.3236 0.5898 1.1842 2.4573 + + + − − 푤푐 푤퐼 < 푤푐 푤푃 < 푤푐 < 푤퐼 푤퐼 < 푤푐 < 푤푃 푤푐 < 푤퐼 E wI −12.5099 −2.4404 −0.6733 −0.2259 −0.0531 0.0799 0.3236 1.1842 + + − − wc wI < wc wP < wc < wI wI < wc < wP wc < wI Example - - Figure3a - Figure3b - Figure3c Figure3d

More importantly, it is shown that Q(w) has a ﬂat magnitude characteristic in both the passband and the stopband when E(wP) in terms of K is negatively minimized, and, + then, wc falls in between wP and wI : + wP < wc < wI (26)

This corresponds to K = 7 in Table1 and, as shown in Figure3b, no overshoot and no undershoot appear in the passband and the stopband. For the further evaluation of this result, in the case of wc = 0.7071 (corresponding to ωc = 0.25π) and N = 28, Figure4a,b show E(w) in terms of K and the effect on Q(w), respectively. The related parameters are also indicated in Table2 where δO and δU are the peak values of overshoot and undershoot in the passband and the stopband, respectively. It can be seen that taking K = 23 yields the smallest negative value E(wP)|K=23 (at wP|K=23 = 0.6429) and results | + = in wP K=23 and wI K=23 satisfying Equation (26) for a given wc 0.7071. Consequently, using K = 23 potentially leads to E(w)|K=23 yielding a cut off-frequency fixing filter with zero overshoot and zero undershoot (i.e., δU = δO = 0). Especially, it is shown that the filters by K = 22 and 24 have a tolerant magnitude distortion, but relatively narrower transition band than the filter by K = 23. + From the results so far, it has been shown that K has to be chosen so that wP and wI of E(w) satisfy the condition of Equation (26) for the prescribed cut off point wc. To determine Electronics 2021, 10, 553 8 of 15

Electronics 2021, 10, x FOR PEER REVIEW 8 of 15

+ an optimal value of K under this condition, substituting wP and wI given by Equations (21) and (23) into Equation (26) and simplifying the algebra, yields: Example - - Figure 3a - p Figure 3b - Figure 3c Figure 3d 2N + (2N − 1)w − 4N + (1 − 4N)w2 N(1 + w ) c c < K < c , for integer K (27) 4 2

: point 1.0 wP Qg()w K=5 x :  point wI  o : wI point 1 (=-3dB) 0.5 2

()w K=5 0.0 Q()w K=5

-0.5 x o wc   -1.0 c

1.0 0.5 0.0 -0.5 -1.0 w  0.0 0.5 1.0

(a)

: point wP 1.0 x :  point Q () wI g w K=7  o : point Q()w K=7 wI 0.8

1 (=-3dB) 0.6 2

0.4 wc c 0.2

()w K=7 0.0 x o

-0.2 w 1.0 0.5 0.0 -0.5 -1.0  0.0 0.5 1.0

(b)

1.4 : point wP x :  point Q()w K=9 wI 1.2  o : wI point

1.0 Qg()w K=9

0.8 1 (=-3dB) 0.6 2

0.4 o x  0.2 wc

c 0.0 ()w K=9 w 1.0 0.5 0.0 -0.5 -1.0  0.0 0.5 1.0

(c)

Figure 3. Cont.

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Electronics 2021, 10, 553 9 of 15

2.5 2.5 : point : point Q()w K=10 wP wP Q()w K=10  x : xpoint :  point wI wI   o : wI opoint : w point 2.0 2.0()w I K=10 ()w K=10

1.5 w  1.5 o oc wc   c c x x 1.0 1.0Q ()w g K=10 Qg()w K=10

1 1 0.5 0.52 2

0.0 0.0 1.0 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 w-1.0 w  0.0 0.0 0.5 0.5 1.0 1.0 

(d) (d)

± Figure 3. 푄 (푤), Ε(푤), and 푄(푤) and the relation between (푤 , 푤(푤 , ±푤, 푤) due , 푤 ) to 퐾 for 푁 = 11± and 푤 = 0.4 (휔 = Figure 푔 3. 푄푔(푤), Ε(푤), and Figure푄(푤) 3.andQ gthe(w) , relationE(w), and betweenQ(w)퐼 and 푃 퐼 the푐 푃 relation푐 due between to 퐾 for w푁I ,=w11P , wandc푐 due 푤푐 to= 0K.푐4for (휔N푐 = 11 and + + + + + − − + − − − 0.3690휋.369): (a휋) )푤: (퐼a)< 푤푤푐

: point wP : w point K=26 K=26  P x : w pointx :  point 1 1 I wI   o : w pointo : point K=24 K=24 I wI K=25 K=25 K=23 KK=22=23 K=22 0 0 K=21 K=21

)

( ( K=20 K=20

 -1  -1 :(w | ,: ((w| | , ))( | )) P K=23 wP KP=23K=23 wP K=23

w  -2 c wc  -2 c c K=19 K=19

-3 -3     w w 1.0 1.0c 0.5c 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0   0.0 0.0 0.33 0.33 0.5 0.5 0.66 0.66 1.0 1.0

(a) (a)

1.8 1.8

1.5 1.5 1.2 K1.2=26 K=26   1.2 1.2 K=25 K=25 o o 0.9 0.9 1.0 1.0 0.6 0.6

0.3 0.3 0.8 0.8 0.0 0.0 1 1 -0.3 2 -0.3 0.6 0.6 2 1.0 0.51.0 0.00.5 -0.50.0 -1.0-0.5 w -1.0 w 0.0 0.0 0.5 0.5 1.0  1.0 



 K=23 0.4  0.4 K=23

Q K=24 K=24 K=22 K=22 0.2 0.2

w  0.0 0.0 c wc ( ( K=21 K=21 c c u u -0.2 -0.2

1.0 1.0 0.5 0.5 0.0 w0.0 w   0.0 0.0 0.25 0.25 0.5 0.5

(b) (b) Ε(푤) Ε(푤) 푄(푤) 퐾 푁 = 28 푤 = 0.7071 FigureFigure 4. (a ) 4. (a) Εand(푤) (andb) the (b ) effect theFigure effect of the 4. of( a related ) theE( w related) and (b Εon)(푤 the ) effecton 푄 due( of푤) the todue related increasing to increasingE(w ) on for Q퐾 ( wfor) due 푁 = toand28 increasing and푐 푤푐K=for0. 7071N = 28 and (휔푐 =(휔0.25=휋0)..25 휋). 푐 wc = 0.7071 (ωc = 0.25π).

Electronics 2021, 10, 553 10 of 15

Table 2. Related parameters for Figure4.

K Parameters 19 20 21 22 23 24 25 26

CK −718.85 −475.26 −293.62 −151.91 −41.13 37.64 80.72 90.53 − wI 0.1741 0.2500 0.3272 0.4058 0.4862 0.5687 0.6539 0.7433 wP 0.3571 0.4286 0.5000 0.5714 0.6429 0.7143 0.7857 0.8571 + wI 0.5274 0.5918 0.6550 0.7166 0.7766 0.8344 0.8894 0.9404 E(wc) −0.2882 −0.2776 −0.2499 −0.1884 −0.0743 0.0991 0.3097 0.5061 − E wI −2.6494 −1.0075 −0.4267 −0.1832 −0.0510 0.0612 0.2316 0.6708 E(wP) −4.3577 −1.6531 −0.6983 −0.2990 −0.0829 0.0993 0.3738 1.0761 + E wI −2.5816 −0.9755 −0.4101 −0.1746 −0.0480 0.0569 0.2106 0.5861 δo 0.0 0.0 0.0 0.0 0.0 0.0058 0.1594 0.8101 Electronicsδ u2021, 10, x FOR−3.7859 PEER REVIEW−1.1088 −0.2241 −0.0021 0.0 0.0 0.011 0.0 of 15 + + − − wc wI < wc wP < wc < wI wI < wc < wP wc < wI

20 EQ.(15)

= 20 - 60 EQ.(28) N 10 Optimal K

20 40 60 80 100 N

(a)

K 40

EQ.(15) 20 = 20 - 60 EQ.(28) N Optimal K

20 40 60 80 100

(b)

Figure 5. Accuracy comparisonFigure of Equation 5. Accuracys (15) comparisonand (28) due of to Equations 푁 for (a) (15)푤푐 = and0.4 (28) (휔푐 due= 0 to.369N휋for) and (a) (wbc) =푤푐0.4= 0(ω.7071c = 0.369 π) 휔 = 0.25휋 ( 푐 ). and (b) wc = 0.7071 (ωc = 0.25π).

Hence, we can infer from Equation (27) that the empirical formula of Equation (15) given by Hermann is a wrong expression since K in accordance with Equation (15) does not satisfy the condition Equation (26) (i.e., wP < wc). From the fact that choosing the

Figure 6. Design procedure of cut off-frequency fixing filters.

Electronics 2021, 10, x FOR PEER REVIEW 11 of 15

30 Electronics 2021, 10, 553 11 of 15 20 EQ.(15)

= 20 - 60 EQ.(28) N 10 Optimal K optimal value of K has to satisfy Equation (27), a new formula that is more accurate than 20 40 60 80 100 Equation (15) can beN deﬁned as: " p # 4N − (1 − 4N)w − 4N + (1 − 4N)w2 (aK) = c c + 0.5 (28) 4

80 by averaging the upper and lower limit values of Equation (27). To verify the effectiveness of this formula, Figure5 shows the accuracy comparison of Equations (15) and (28) for wc = 0.4 (ωc = 0.369π) and wc = 0.7071 (ωc = 0.25π) where the “F” symbol denotes the 60 optimal K chosen by negatively minimizing E(wP) under condition (27). This indicates that Ks obtained according to Equation (28) are remarkably consistent with the optimal Ks. K Based on results so far, Figure6 exhibits a design procedure to directly compute the 40 coefficients of cut off-frequency fixing filters with a desired frequency response. Table3 and Figure7 show the comparison between the proposed method and existing methods [ 15,23] for N = 20 and 45. From Table3 EQ.(15), it can be said that the proposed filters apparently are 20 EQ.(28) = 20 - 60 −14 a “cut off errorN free” and “flat magnitude” Optimal K since δo < 10 can be very negligible and ignored. On the other hand, previous filters using empirical formula of Equation (15) are maximally20 flat40 in the passband60 80 and stopband,100 while there exists a filter cut off-frequency error of 2.2% to 6.6%.N In addition, it appears that the iteration training method [23] leads to very slight cut off errors despite performing multiple computations to determine K, as shown in Figure(b 4) in Reference [23]. Consequently, these examples demonstrate that the proposed method derives accurate FIR filters with a magnitude response while allowing Figure 5. Accuracy comparison of Equations (15) and (28) due to 푁 for (a) 푤푐 = 0.4 (휔푐 = 0.369휋) and (b) 푤푐 = 0.7071 (휔푐 = 0.25휋). direct and simple computation in designing such a filter.

Figure 6. Design procedure of cut off-frequency ﬁxing ﬁlters.

Figure 6. Design procedureFigure8, of using cut off Figure-frequency6, shows fixing filters. cut off-frequency fixing filters for ωc = 0.369π (wc = 0.4) and various N, and the related values are given in Table4. Table5 also indicates coefficients of the cut off-frequency fixing filter for (N, K) = (20, 13), where gî and hn have been obtained according to Equations (20) and (14), respectively.

Electronics 2021, 10, x FOR PEER REVIEW 12 of 15

Electronics 2021, 10, 553Table 3. Comparison of three design methods for (a) 푁 = 20 and (b) 푁 = 45. 12 of 15

흎풄 = ퟎ. ퟑퟔퟗ흅 (풘풄 = ퟎ. ퟒ) 흎풄 = ퟎ. ퟐퟓ흅 (풘풄 = ퟎ. ퟕퟎퟕퟏ) MAXFLAT Cutoff Fixing MAXFLAT Cutoff Fixing Parameters FiltersTable of 3. ComparisonReference of[23 three] designFilters methodsof Filters for (a) ofN = 20Reference and (b) N =[2345.] Filters of Equation (10) Equation (18) Equation (10) Equation (18) w w 1 ωc = 0.369π ( c = 0.4) ωc = 0.25π ( c = 0.7071) 휔3푑퐵 0.3502 MAXFLAT0.3690 0.3690 Cutoff0. Fixing2419 MAXFLAT0.25 0.25 Cutoff Fixing Parameters −5 (휔푐−휔3푑퐵)/π 0.0188 Filters2.14 of × 10 Reference0.0 [23] Filters0.0081 of Filters of0.0 Reference [230.0] Filters of 3 Equation (10) Equation (18) Equation (10) Equation (18) 훿표 1 0.0 0.0 0.0 0.0 0.0 0.0 ω3dB 0.3502π 0.3690π 0.3690π 0.2419π 0.25π 0.25π 훿푢 0.02 0.0 −0.05 0.0 0.0 0.0 푁 = 20 (ωc − ω3dB)/π 0.0188 2.14 × 10 0.0 0.0081 0.0 0.0 δ 3 0.0training 0.0 0.0 0.0training 0.0 0.0 K decision oEquation (15) Equation (28) Equation (15) Equation (28) N = 20 δu (130.0 iteration) 0.0 0.0 (160.0 iterations) 0.0 0.0 training training accuracy/ K decision Equation (15) Equation (28) Equation (15) Equation (28) (13 iteration) (16 iterations) 4 low/low high/ very high high/low low/low high/ very high high/low complexity accuracy/ high/very high/very 1 4 low/low high/low low/low high/low 휔3푑퐵 complexity0.3445 0.3690 high0.3690 0.2445 0.25 high 0.25 1 ω 0.3445π −7 0.3690π 0.3690π 0.2445π −12 0.25π 0.25π (휔푐−휔3푑퐵)/ 3dB 0.0245 3.14 × 10 0.0 0.0055 1.83 × 10 0.0 −7 −12 3 (ωc − ω3dB)/ π 0.0245 3.14 × 10 −15 0.0 0.0055 1.83 × 10 −16 0.0 훿표 3 0.0 0.0 3.11× 10 0.0−15 0.0 4.44 × 10 −16 δo 0.0 0.0 3.11× 10 0.0 0.0 4.44 × 10 훿푢 0.0 0.0 0.0 0.0 0.0 0.0 푁 = 45 N = 45 δu 0.0 0.0 0.0 0.0 0.0 0.0 training training training training K decision K decisionEquation (15) Equation (15) Equation (28)Equation Equation (28) (15) Equation (15) Equation (28)Equation (28) (30 iteration)(30 iteration) (37 iterations)(37 iterations) accuracy/ high/very high/very low/low high/low low/low high/low accuracy/complexity 4 high high 4 low/low high/ very high high/low low/low high/ very high high/low 1 complexity 2 3 ω3dB denotes the actual cut off-frequency of the ﬁlter. (ωc − ω3dB)/π indicates the cut off-frequency error of the ﬁlter. δo was calculated 1 2 휔3푑퐵 denotesto sixteen the decimal actual places. cut off4 this-frequency denotes Kofdecision the filter accuracy. (휔푐− and휔3 computation푑퐵)/π indicates complexity. the cut off-frequency error of the filter. 3 4 훿표 was calculated to sixteen decimal places. this denotes K decision accuracy and computation complexity.

1.000002

1.0 1.000000

0.999998 0.8 1 2 0.999996 (=-3dB) 0.00 0.05 0.10 0.6

0.4 Herrmann's design Proposed method Ref.[23] 0.2   c 0.0

0.0 0.2 0.4 0.6 0.8 1.0 

(a)

1.0000002 1.0 1.0000000

0.8 1 0.9999998 2 (=-3dB) 0.00 0.05 0.10 0.15 0.6

0.4 Herrmann's design Proposed method Ref.[23] 0.2   c 0.0

0.0 0.2 0.4 0.6 0.8 1.0 

(b)

Figure 7. Comparison of three design methods: (ωc = 0.369π) for (a) N = 20 and (b) N = 45.

Electronics 2021, 10, x FOR PEER REVIEW 13 of 15

Electronics 2021, 10, 553 13 of 15 Figure 7. Comparison of three design methods: (휔푐 = 0.369휋) for (a) 푁 = 20 and (b) 푁 = 45.

(63, 42) 1.0

0.8

1 (=-3dB) 2  0.6

cos (45, 30)

 (28, 18)



w 0.4  (20, 13)

Q (14, 9) 0.2 (10,6)   (5, 3) c 0.0

0.0 0.2 0.4 0.6 0.8 1.0 

Figure 8. Cut off-frequency ﬁxing ﬁlters with various N and ωc = 0.369π (wc = 0.4). Figure 8. Cut off-frequency fixing filters with various 푁 and 휔푐 = 0.369휋 (푤푐 = 0.4). Table 4. Related parameters of Figure8. Table 4. Related parameters of Figure 8.

NN ParametersParameters 1010 1414 2020 28 45 45 63 63 퐾K 6 6 99 1313 18 30 30 42 42 CK −0.5846 −0.7311 −1.8770 −10.073 −112.62 −4055.5 퐶퐾 −0.5846 −0.7311 −1.8770 −10.073 −112.62 −4055.5 wP 0.2000 0.2857 0.3000 0.2857 0.3333 0.3333 w푤+푃 0.50000.2000 0.53180.2857 0.50590.3000 0.2857 0.4618 0.3333 0.4702 0.3333 0.4495 I+ 푤퐼 0.5000 0.5318 0.5059 0.4618 0.4702 0.4495

TableTable 5. 5. CoefficientsCoefficients of of the the cut cut off off-frequency-frequency fixing fixing filter filter for for ((푁N,,퐾K))==((2020,, 13 13).) .

gi 푔푖 hnℎ푛 푔0 = −1.198409231 h0 = −0.000000018 h11 = −0.003859813 g0 = −1.198409231 h0 = −0.000000018 h11 = −0.003859813 g1푔1= = 21.309955797 21.309955797 h1 =h1− =0.000000147 −0.000000147 h12 == −−0.0054508690.005450869 g2푔=2 =− 169.484927527−169.484927527 h2 =h2− =0.000000073 −0.000000073 hh1313 == 0.005958058 0.005958058 g3 = 802.886975733 h3 = 0.000003072 h14 = 0.020963265 푔3 = 802.886975733 h3 = 0.000003072 h14 = 0.020963265 g4 = −2532.615452798 h4 = 0.000011169 h15 = 0.004171941 푔4 = −2532.615452798 h4 = 0.000011169 h15 = 0.004171941 g5 = 5615.229929033 h5 = −0.000003022 h16 = −0.048651840 g6푔=5− = 8988.1354208935615.229929033 h6 =h5− =0.000097254 −0.000003022 h1176 == −−0.0486518400.055049249 g7푔=6 = 10,505.014699770 −8988.135420893 h7 =h6− =0.000159706 −0.000097254 hh1718 = =−0.055049249 0.078213381 h = 0.000280582 h = 0.297774354 푔7 = 10,505.014699770 8 h7 = −0.000159706 h1198 = 0.078213381 h9 = 0.001164512 h20 = 0.408751644 h8 = 0.000280582 h19 = 0.297774354 h10 = 0.000355835 All coefficients were rounded off to nine decimal places.h9 = 0.001164512 h20 = 0.408751644 h10 = 0.000355835 All4. coefficients Conclusions were rounded off to nine decimal places. Problems with the cut off-frequency error always arise in a MAXFLAT filter design. 4. Conclusions This is due to the fact that the maximum possible number of zeros at z = ±1 is imposed, whichProblems leaves nowith degree the cut of freedom,off-frequency and, thus,error noalways independent arise in a parameters. MAXFLAT In filter this design. paper, a Thisnew is method due to the has fact been that proposed the maximum providing possible a cut off-frequency number of zeros error at compensation 푧 = ±1 is imposed, function, whichavailable leaves to previousno degree closed-form of freedom, polynomials, and, thus, no for independent compensating parameters. for the cut In off-frequency this paper, a errornew method of MAXFLAT has bee FIRn proposed filters. It providing has been a shown cut off that-frequency this error error compensation compensation function function,derives available a closed-form to previous Chebyshev closed-form polynomial polynomials, characterizing for compensating cut off fixing forFIR the filterscut off with-fre- a quencyprescribed error cut of off-frequencyMAXFLAT FIR and filters. allows It ahas computationally been shown that efficient this anderror accurate compensation formula functionto obtain derives such filters a closed with-form flat magnitudeChebyshev responsepolynomial characteristics characterizing for cut a given off fixing order FIR of a filtersfilter with and cuta prescribed off-frequency. cut off The-frequency examples and were allows shown a tocomputationally provide a complete efficient and and accurate ac- curatesolution formula for the to design obtain of such such filters filters. with Hence, flat a magnitude solution to theresponse problem characteristics encountered for in thea previous methods is found.

Electronics 2021, 10, 553 14 of 15

Author Contributions: Conceptualization, D.C., W.C. and J.J. Methodology, D.C. Software, D.C. Vali- dation, D.C., W.C. and I.J. Formal analysis, D.C. and J.J. Investigation, D.C., W.C. and I.J. Resources, J.J. Data curation, W.C. and I.J. Writing—original draft preparation, D.C. and J.J. Writing—review and editing, D.C. Visualization, D.C. Supervision, J.J. Project administration, J.J. Funding acquisition, J.J. All authors have read and agreed to the published version of the manuscript. Funding: This work was funded by the Korea Institute of Energy Technology Evaluation and Plan- ning (no. 20194030202320) and Korea Evaluation Institute of Industrial Technology (no. 20012884), which are funded by the Ministry of Trade, Industry & Energy of the Republic of Korea. Data Availability Statement: Data is contained within the article. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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