- Email: [email protected]

Selectivity nomographs for classical ﬁlters Claude S. Lindquista,1, Celestino A. Corralb,* b

a University of Miami, P.O. Box 248294, Coral Gables, FL 33124, USA Motorola Inc., 1500 Gateway Boulevard, M/S 71, Boynton Beach, FL 33426, USA

Abstract Nomographs for determining the ﬁlter order of classical ﬁlters based on selectivity requirements are presented. The selectivities for a variety of standard classical ﬁlters are summarized in equation form and the general selectivity nomograph is constructed. The selectivity equations are then converted into nomograph form by applying the relationship between the transfer function and the response slope. Design examples are presented to demonstrate the usefulness of the selectivity nomographs. These nomographs can be used to gauge ﬁlter performance and combined with optimization techniques can yield superior classical ﬁlter designs. r 2002 Published by Elsevier Science Ltd. on behalf of The Franklin Institute.

1. Introduction Nomographs are effective ﬁlter design tools. They represent equations in graphical form and therefore give increased insight and understanding in the design process. First, applied to ﬁlter design by Kawakami [1], nomographs reached their peak in the 1960s and 1970s with maturing synthesis techniques [2,3]. The resulting nomographs capture gain information relating passband magnitude ripple Mp ; stopband magnitude ripple/attenuation Ms ; passband and stopband frequencies fp and fs ; respectively, and order n [4]. Nomographs of Butterworth, Chebyshev, elliptic and Bessel ﬁlters are shown in Fig. 1. With the advent of sophisticated ﬁlter design applications such as FILSYN [5] and others, nomographs have gone the way of the slide rule. However, recent works have reawakened an interest in this type of design tool. In [6] a single curve nomograph is

*Corresponding author. Tel.: +1-561-739-2599. E-mail addresses: [email protected] (C.S. Lindquist), [email protected] (C.A. Corral). 1 Present address: 686 Paradise Park Club, Santa Clara, CA 95060, USA. 0016-0032/02/$22.00 r 2002 Published by Elsevier Science Ltd. on behalf of The Franklin Institute. PII: S 0 0 1 6 - 0 0 3 2 ( 0 2 ) 0 0 0 1 5 - 7

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Fig. 1. Gain nomographs for (a) Bessel, (b) Butterworth, (c) Chebyshev and (d) elliptic ﬁlters.

proposed relating Butterworth, Chebyshev and elliptic ﬁlters. More recently, the authors proposed a technique for optimizing classical ﬁlter response in [7] by using the nomograph structure proposed in [8]. In addition, nomographs for transitional ﬁlters have recently been advanced in [9]. Filter selectivity is a measure of the slope of the magnitude response at the ﬁlter’s corner frequency. Maximizing selectivity increases stopband rejection near the passband edge; the transition bandwidth is also minimized thereby reducing ﬁlter noise energy [10,11]. High selectivity bandpass ﬁlters can reduce VCO jitter [12]. Filter selectivity can also pay a role in improving inter-symbol interference (ISI) in partial response data transmission ﬁlters [13].

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Analytically, ﬁlter selectivity provides information on frequency requirements that can be used to modify ﬁlter response [14]. By incorporating selectivity requirements in the approximation formula, the inverse Chebyshev ﬁlter was modiﬁed in [15] with nominal impact on implementation. Following a similar approach, ﬂat delay ﬁlters employ transmission zeros in the stopband to improve band-edge selectivity [16]. In [17] elliptic ﬁlter selectivity was treated as a design goal noting that inﬁnite selectivity was not a barrier to realization. However, when employing selectivity requirements, implementation issues must be carefully considered. High selectivity active RC ﬁlters require higher Q circuits with pole enhancement as shown in [18]. On the other hand, in certain novel designs selectivity can be controlled independently of cut-off frequency [19]. High selectivity requirements can also make demands on digital ﬁlters due to the difﬁculty of achieving steep roll-offs in certain forms [20]. Due to the importance of selectivity in ﬁlter designs, this paper presents nomographs that employ the selectivity criterion in the determination of ﬁlter order. First, we summarize the selectivities in equation form for a variety of standard classical ﬁlters. We next construct the general selectivity nomograph. The selectivity equations are converted into nomograph form for the stated ﬁlter types. The resulting nomographs shall be compared to demonstrate their usefulness and design examples submitted and discussed. By employing the proposed nomographs it is possible to use selectivity directly in the design/synthesis process for classical ﬁlters.

2. Selectivity formulation Classical ﬁlters have gain represented by the relation jHðjoÞj2 ¼

1 ; 1 þ e2 Fn2 ðoÞ

ð1Þ

where e is the ripple parameter and the ﬁlter generating polynomial Fn ðoÞ of order n satisﬁes Fn ð1Þ ¼ 1: The ﬁlter band-edge selectivity BES is deﬁned as the slope of the band-edge gain (1/rad/s) as shown in Fig. 2(a). From Eq. (1), the selectivity equals djHðjoÞj e2 ¼ F 0 ð1Þ: ð2Þ BES ¼ do o¼1 ð1 þ e2 Þ3=2 n The gain polynomials and band-edge selectivities of some standard classical ﬁlters are listed in Table 1. The ﬁlters are ordered in terms of increasing selectivity. Conventionally, the passband ripple factor Mp is deﬁned as the loss at the normalized passband corner frequency o ¼ 1: Therefore, from Eq. (1) 1 1 ¼ Mp2 1 þ e2 Fn2 ð1Þ

ð3Þ

so solving for Mp gives Mp2 ¼ 1 þ e2 :

ð4Þ

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Fig. 2. Selectivity using (a) gain slope BES and (b) Bode gain slope BES dB.

Table 1 Filter selectivities Filter approximation

Selectivity

Bessel

0.491

Butterworth

n pﬃﬃﬃ 2 2 n pﬃﬃﬃ 2 2ð1 b2 Þ n ð1 b2 Þn1 pﬃﬃﬃ b=shaping parameter n 2 2n 2 2 ð1 b Þ þ b

MFMBO n odd

MFMBO n even

e21 n2

Chebyshev

ð1 þ e21 Þ3=2 e21 n

Ultraspherical

ð1 þ

1þ

e21 Þ3=2

n1 a=selectivity parameter 2ða þ 1Þ

0:0884ðn þ 1Þ2 0:0884nðn þ 2Þ

Papoulis n odd Papoulis n even

e21 n2

Elliptic

ð1 þ e21 Þ3=2

1 m0 ; 1m

m ¼ 1=o2s ; m0 ¼ ðe1 =e2 Þ2

Therefore, a ﬁlter’s selectivity can alternately be expressed as

BES ¼

Mp2 1 0 Fn ð1Þ: Mp3

ð5Þ

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This gives a functional relationship between the selectivity BES and its parameters which are the passband ripple Mp and the derivative of the ﬁlter polynomial Fn : 3. General selectivity nomograph The general selectivity nomograph is constructed from Eq. (5). We ﬁrst convert Eq. (5) into a linear equation by taking the log of both sides to obtain ! Mp2 1 logðBESÞ ¼ log ð6Þ þ logðFn0 ð1ÞÞ: Mp3 This has an analogous form to the standard ﬁlter gain nomograph equation involving Mp ; Ms and Fn ðOÞ; where [8] logðMs2 1Þ ¼ logðMp2 1Þ þ logðFn2 ðOÞÞ:

ð7Þ

This is a four-parameter equation involving (Mp ; Ms ; Fn ; O ¼ os =op ). The BES equation is a three-parameter equation involving (BES, Mp ; Fn0 ). We now continue the analysis with the frequency parameter o: In the four-parameter case, we began by deﬁning the g-axis as gn ¼ logðFn2 ðoÞÞ: Taking the derivative of Eq. (8) at the passband corner frequency gives dgn 2Fn Fn0 ¼ ¼ 2Fn0 ð1Þ; do o¼1 Fn2 o¼1 dgn dgn ¼ o ¼ 2Fn0 ð1Þ: d log oo¼1 do o¼1

ð8Þ

ð9Þ

We see that one-half of the initial slope of gn in the gain nomograph produces the graphical value Fn0 ð1Þ: Therefore, we can produce the selectivity nomograph of Eq. (6) using the existing nomograph represented by Eq. (7). We shall maintain the same Mp -axis, utilize the Fn -axis as the Fn0 -axis, and modify the Ms axis to become the BES-axis. We shall now explore their relations including their scaling.

4. Nomograph of selectivity from gain We have from Eq. (9) that Fn0 ð1Þ equals 1 dgn 1 d log Fn2 0 ¼ : Fn ð1Þ ¼ 2 d log oo¼1 2 d log o o¼1

ð10Þ

As mentioned earlier, it equals one-half the slope of the nth-order ﬁlter polynomial squared Fn2 evaluated at o ¼ 1 when plotted on log–log paper. By drawing a tangent line the nth-order curve (denoted as gnt ) at o ¼ 1 in the gain nomograph and noting its intersection point gnt ð10Þ on the right-hand g-axis (at o ¼ 10), we graphically

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Fig. 3. Gain nomograph construction to estimate selectivity.

determined the Fn0 ð1Þ value to equal pﬃﬃﬃﬃﬃ Fn0 ð1Þ ¼ 12 gnt ð10Þ ¼ gnt ð 10Þ:

ð11Þ

Because pﬃﬃﬃﬃﬃ the tangent line gnt follows a power law, half its value at 10 equals its value at 10: This construction is shown in Fig. 3. Substituting this result into Eq. (5) gives pﬃﬃﬃﬃﬃ Mp2 1 gnt ð10Þ Mp2 1 ¼ BES ¼ gnt ð 10Þ: ð12Þ 3 3 2 Mp Mp Now consider the ripple factor RF ¼ ðMp2 1Þ=Mp3 in Eqs. (5) and (12) which is plotted in Fig. pﬃﬃ4.ﬃ Solving dðRF Þ=dMp ¼ 0 shows that the maximum RF equals 0.385 when Mp ¼ 3 ¼ 2:38 dB: For large ripples, Mp b1 and RF E1=Mp 51: For small ripples, Mp E1 and RF EMp2 1E0: Noting that ax 1Ex ln a for ax E1 and taking Mp2 ¼ 10Mp ðdBÞ=10 ; then RF EMp ðdBÞ lnð10Þ=10 so a good approximation is RF EMp ðdBÞ=4:4: As a practical matter, passband ripple seldom exceeds 3 dB which pﬃﬃis ﬃ considered large. a passband gain ripple of 3 dB corresponding to Mp ¼ 2; then RF ¼ pﬃﬃFor ﬃ 1=2 2: For small passband ripple of 0.1 dB where Mp ¼ 1:012; then RF ¼ 0:0225 and the ripple factor becomes very small. Multiplying the slope Fn0 ð1Þ and the ripple factor RF gives the band-edge selectivity of Eqs. (5) and (12). Since 0oRF o0:385; the maximum selectivity equals 0:385Fn0 ð1Þ: To demonstrate the effectiveness of the approach, consider the Butterworth ﬁlter where gn ð10Þ ¼ 2n from the gain nomographpin ﬃﬃﬃ Fig. 1(b). From Eq. pﬃﬃﬃ (11) the slope Fn0 ð1Þ ¼ n: Using the 3 dB ripple factor 1=2 2; then BES ¼ n=2 2 from Eq. (5).

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Fig. 4. Ripple factor ðRF Þ ¼ ðMp2 1Þ=Mp3 for (a) numeric and (b) dB value of Mp :

This matches the analytical result from Table 1. As another example, consider the Bessel ﬁlter whose gain magnitudes are all tangent to the g-curve at o ¼ 1 with the same slope as Fig. 1(a). This tangent line lies between the n ¼ 1 2 curves and pﬃﬃﬃ intersects the right-hand axis between 2 and 3.6 (slope 1–1.8). Using gnt ð10ÞE2 2 pﬃﬃﬃ and RF ¼ 1=2 2; then BESE12; which also agrees with Table 1. As a ﬁnal example, notice that the Chebyshev and elliptic ﬁlters that have gain nomographs (Fig. 1(c) and (d)) curving concave upward rather than downward like the Bessel ﬁlters. Therefore, their tangent lines from the origin have higher slopes than the Butterworth and much higher slopes than the Bessel ﬁlters. For example, the n ¼ 3 elliptic pﬃﬃﬃ ﬁlter has g3t ð10ÞE20 so its band-edge selectivity is about 3 for RF ¼ 1=2 2:

5. Transition bandwidth and selectivity As mentioned previously, selectivity gives an indication of the transition bandwidth from the passband to the stopband of the ﬁlter. The approximate transition bandwidth can be estimated as follows. The band-edge selectivity in dB of the Bode gain slope of the ﬁlter is denoted as BESdB equaling d logjHðjoÞj BESdB ¼ 20 d log o o¼1 o djHðjoÞj ¼ 20 jHðjoÞj do o¼1 ¼ 20Mp BES dB=dec:

ð13Þ

The selectivity is shown in Fig. 2(b). The BES relation is obtained by evaluating the slope using Eq. (2) and the gain jHðj1Þj using Eqs. (1) and (3). Notice that BES has the units of 1/rad/s and multiplying it by 20Mp expresses BES in units of dB/dec.

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Mp BES is the power law slope. The derivative in Eq. (13) can be approximated using differences as d logjHðjoÞj logjHs j logjHp j logðMs =Mp Þ ¼ 20 E ¼ 20 ; ð14Þ 20 d log o log o logð1Þ log o t

o¼1

t

where Ms is the minimum stopband attenuation and ot is the approximate transition frequency. Combining Eqs. (13) and (14) and solving for ot gives ot EðMs =Mp Þ1=Mp BES ;

Mp BESb1:

ð15Þ

The transition bandwidth is then ot 1 ¼ logðMs =Mp Þ=ðMp BESÞ:

ð16Þ

Ideally, ot would equal the ﬁlter stopband frequency os ; but this is seldom the case, hence necessitating the use of gain nomographs. Usually, ot oos in high selectivity ﬁlters such as Chebyshev and elliptic ﬁlters, and ot > os in low selectivity ﬁlters such as the Bessel approximation. They are about equal in compromise ﬁlters like the Butterworth response (Fig. 5). 6. Selectivity nomographs of classical ﬁlters Having laid the groundwork for the selectivity relations, we now generate the selectivity nomographs. As discussed in Section 3, this nomograph is a pictorial representation of Eq. (6), having three parameters (BES, Mp ; Fn0 ð1Þ). We begin by considering Fn0 ð1Þ: In general, the derivative of the ﬁlter gain polynomial evaluated at the corner frequency o ¼ 1 is ð1 þ e2 Þ3=2 BES ð17Þ e2 which is a reformulation of Eq. (2) where the BES are listed in Table 1. Performing this multiplication gives the different Fn0 ð1Þ also listed in Table 1. For example, the band-edge derivative of the ultraspherical ﬁlter is one of the most general Fn0 ð1Þ ¼

Fig. 5. Transition bandwidth of (a) Chebyshev and (b) Bessel ﬁlters.

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forms having

n1 Fna ð1Þ ¼ n 1 þ : 2ða þ 1Þ

ð18Þ

Setting a ¼ N; 12; 0, and 12 yield the derivatives for the Butterworth, 2nd Chebyshev, Legendre, and 1st Chebyshev ﬁlters, respectively. This is easily veriﬁed by the table. From Eq. (6), the general selectivity equation has the form ! Mp2 1 log BES log ð19Þ ¼ logðFn0 ð1ÞÞ: Mp3 This is a linear equation having the form x y ¼ a ¼ x ða xÞ: This shows that the selectivity BES and Mp must balance themselves according to Eq. (19) so that the difference is constant. Let us consider each term individually. We see from Table 1 that 1oFn0 ð1Þon2 at least. Selecting the maximum ﬁlter order to be considered as 30, then Fn0 ð1ÞAð1; 900Þ at least. Therefore, we choose 0ologðFn0 ð1ÞÞo3 as the extended range of the right-hand side of Eq. (19). The maximum passband ripple Mp may pﬃﬃﬃ be reasonably bounded by 0:001 dBoMp ðdBÞo3 dB; so 1:00012oMp o 2: Therefore, 0:00023oðMp2 1Þ= Mp3 o0:354 and 3:64ologððMp2 1Þ=Mp2 Þo 0:45: We choose (3.64, 0.45) as the range of the middle term in Eq. (19). Finally, from Table 1, 0:5oBESon2 : Therefore, we select 0olog BESo3: This is the range of the left-hand term of Eq. (19). We now construct the nomograph spacings and scales following the method described in [8]. The three vertical lines are chosen to be the (Mp ; BES; Fn0 ð1Þ) variables taken in that order from Eq. (19). We chose to space the lines as ð0; 12; 1Þ as is done in the gain nomograph. Because Eq. (18) has the linear from x y ¼ a; the nomograph construction must satisfy the determinant 0 logððMp2 1Þ=Mp3 Þ 1 ð20Þ 1=2 ð1=2Þlog BES 1 ¼ 0: 1 0 logðFn ð1ÞÞ 1 The ﬁrst row requires that at location 0 with logððMp2 1Þ=Mp3 Þ scale we get unity. The second row requires that at location 12 (halfway between location 0 and 1) with ð12Þ log BES scale we get unity. The third row requires that at location 1 with logðFn0 ð1ÞÞ scale we get unity. Solving Eq. (20) yields log BES ¼ logðFn0 ð1ÞÞ logððMp2 1Þ=Mp3 Þ:

ð21Þ

From the basic determinant of Eq. (2), the (x, y) coordinate axes satisfy ! Mp2 1 þ c1 ; x1 ¼ 0; y1 ¼ log Mp3 x2 ¼ 12;

y2 ¼ 12 log BES þ c2 ;

x3 ¼ 1;

y3 ¼ logðFn0 ð1ÞÞ þ c3 ;

ð22Þ

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where an offset constant ci is introduced into each equation of Eq. (22). As described above, we decided to use Mp ; BES Fn0 ð1Þ within the deﬁned ranges. We select the maximum values to lie around 3 cm on the three support scales. We select the minimum values to lie at 0 cm. Setting the minimum values (0.001, 1, 0.001) in Eq. (22), we ﬁnd that the minimum axes locations are 1:0001152 1 þ 3:64 ¼ 0; 1:0001153

x1 ¼ 0;

y1 ¼ log

x2 ¼ 12;

y2 ¼ 12 logð0:001Þ þ 1:5 ¼ 0;

x3 ¼ 1;

y3 ¼ logð1Þ þ 0 ¼ 0:

ð23Þ

Mp ¼ 0:001 dB corresponds to a numerical value 100.001/20=1.000115. Offset constants of (c1 ; c2 ; c3 ) of 3.638, 1.5 and 0, respectively, set each yi ¼ 0 as desired. Evaluating Eq. (22) using these offsets at the maximum parameter values of (3, 1000, 1000) give the maximum axes locations of 1:4142 1 þ 3:64 ¼ 3:19; 1:4143

x1 ¼ 0;

y1 ¼ log

x2 ¼ 12;

y2 ¼ 12 logð1000Þ þ 1:5 ¼ 3;

x3 ¼ 1;

y3 ¼ logð1000Þ þ 0 ¼ 3;

ð24Þ

where Mp ¼ 3 dB corresponds to a numerical value 103/20=1.414. This veriﬁes the variable ranges that are being used properly span the axes. Now that the selectivity nomograph axes have been constructed, it remains to simply plot the band-edge slope Fn0 ð1Þ for the various ﬁlters from Table 1 at selected orders. To conserve space and make comparison easier, we plot all the slopes on a single graph somewhat analogous to what is done on the gain nomographs. Each ﬁlter is referred to by a reference number k which is plotted horizontally. Its slopes are then plotted vertically as Fn0 ð1Þ for the various orders. Thus the kth vertical line has points (k; Fn0 ð1Þ) plotted along its length. We begin with the lowest selectivity ﬁlter in Table 1 which is Bessel. The next ﬁlters in order are Butterworth, MFMBO, 2nd Chebyshev, Legendre, Papoulis, 1st Chebyshev, and ultraspherical. The highest selectivity in Table 1 belongs to elliptic ﬁlters. To show ﬁlter capabilities more clearly, we have used orders 1–10. This is somewhat less than the range we discussed earlier but the reader will see our reasoning momentarily.

7. Illustrative examples Let us summarize the nomographs we have available at this point. Various ﬁlter gain nomographs are shown in Fig. 1. Filter selectivities were constructed on these gain nomographs in Fig. 6. New ﬁlter selectivity nomographs are shown in Figs. 7 and 8. Although it is extremely useful to be able to evaluate ﬁlter sensitivity directly on a gain nomograph, it is somewhat cumbersome to compare the sensitivities of

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Fig. 6. Selectivity estimates using gain nomographs for (a) Bessel, (b) Butterworth, (c) Chebyshev and (d) elliptic ﬁlters.

various ﬁlters using this method. Indeed, one of the greatest utilities of gain nomographs is to quickly determine ﬁlter order. Once the (Mp ; Ms ) values are entered on the vertical axes, g is determined. Then, locating the (g; Os ) point in the graphical portion determines order by ﬁnding the ﬁrst curve that lies above this point. To determine the order of the ﬁlter types, we simply enter this same (g; Os ) point on the other nomographs. No further construction is necessary. In this sense, the selectivity nomographs are useful. Notice that connecting the equal-order points together from left to right of Fig. 8 determines 10 curves. These curves are roughly straight lines on semilog paper.

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Fig. 7. (a) Selectivity nomograph and (b) gain nomograph construction.

Fig. 8. Selectivity nomographs for (1) Bessel, (2) Butterworth, (3) MFMBO, (4) 2nd Chebyshev, (5) Legendre, (6) Papoulis, (7) 1st Chebyshev, (8) 0.9 ultraspherical, (9) 0.95 ultraspherical and (10) elliptic ﬁlters of orders 1–10.

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Therefore, the selectivity increases exponentially with these ﬁlter types for ﬁxed ﬁlter order. This simple fact cannot be so evidently established using the construction of Fig. 6. Since the nomograph relates three parameters (Mp ; Ms ; n), we can ﬁx one of these parameters and obtain two degrees of freedom in the design (Fig. 9). These are three combinations or design possibilities which are shown in Figs. 10–12. We ﬁx ﬁlter order and type in Fig. 10, ﬁx selectivity in Fig. 11, and ﬁx passband ripple in Fig. 12. We will now give an example of each situation and discuss the ﬂexibilities obtained. Fix filter type and order: Fig. 10 We choose a Papoulis ﬁlter of order 10. From Fig. 8, it has g ¼ 60: Entering 60 on the axis in Fig. 7(b) in the manner shown in Fig. 10 shows that there are an inﬁnite number of (Mp ; BES) combinations. We could specify Mp ¼ 0:18 dB and obtain BES=10. Or we might reduce the passband ripple to Mp ¼ 0:1 dB. and obtain BES=8. Fix selectivity: Fig. 11 Instead let us specify a band-edge selectivity BES=1. Using the construction of Fig. 11, then there are an inﬁnite number of possibilities for (Mp ; Fn0 ð1Þ). For example, using 0.1 dB gives Fn0 ð1Þ ¼ 10: Instead, increasing passband ripple to 0.5 dB shows Fn0 ð1Þ ¼ 2: This gives a range of ﬁlter orders from n1 ¼ 1 to n2 ¼ 3 for Papoulis ﬁlters. Fix passband ripple: Fig. 12 Finally, let us ﬁx the passband ripple as Mp ¼ 0:1 dB. Then from Fig. 12, for BES=1 with Fn0 ð1Þ ¼ 30: Instead, if BES=10 we ﬁnd Fn0 ð1Þ ¼ 300: For the Papoulis ﬁlter, this shows a range of ﬁlter orders from n1 ¼ 3 to well over n ¼ 10: These examples demonstrate how useful design nomographs can be. They eliminate the need for the designer to evaluate equations and instead allow him/her to utilize the nomographs. Not only does this speed up the design process but it shows interrelationships between ﬁlters that are not obvious from their transfer functions. Despite the availability of sophisticated ﬁlter design applications,

Fig. 9. Determining ﬁlter type and order using selectivity nomograph.

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Fig. 10. Trade-off between Mp and BES for ﬁxed ﬁlter type and order.

Fig. 11. Trade-off between in-band gain Mp and order n for ﬁxed selectivity.

Fig. 12. Trade-off between band-edge selectivity BES and order n for ﬁxed in-band ripple.

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nomographs still provide an intuitive means to obtain key information from the captured graphical relationships.

8. Conclusion In the past, only ﬁlter gain nomographs have been available. We have presented a new set of ﬁlter nomographs which describe the relationship between ﬁlter parameters and band-edge selectivity. We have shown how ﬁlter selectivity can be determined directly from a ﬁlter gain nomograph. It was shown that although this is useful, it is not the most convenient form to use. Thus, we developed these new selectivity nomographs. We have brieﬂy reviewed both the basis and construction of general ﬁlter nomographs. An analogy was established between the ﬁlter gain and the ﬁlter selectivity equations. Since they had a similar form, we could transfer the concepts learned in forming gain nomographs to generate the new selectivity nomographs. A representative group of classical ﬁlters was considered and their selectivity nomographs constructed. To conserve space and accelerate the design process, these different selectivity nomographs were condensed into one nomograph. We made the useful observation that the selectivity increased exponentially with ﬁlter type when they were ordered in terms of decreasing transition bandwidth (i.e., therefore more selective).

References [1] M. Kawakami, Nomographs for Butterworth and Chebyshev ﬁlters, IEEE Trans. Circuit Theory CT10 (1963) 288–289. [2] A.I. Zverev, Handbook of Filter Synthesis, Wiley, New York, 1967. [3] G.E. Hansell, Filter Design and Evaluation, Van Nostrand Reinhold, New York, 1969. [4] C.S. Lindquist, Active Network Design with Signal Filtering Applications, Steward & Sons, Long Beach, CA, 1977. [5] FILSYN is available from ALK Engineering, URL: http://www.web-span.com/alk/. [6] P.M. Lin, Single curve for determining the order of an elliptic ﬁlter, IEEE Trans. Circuits Syst. 37 (1990). [7] C.A. Corral, C.S. Lindquist, Design for optimum classical ﬁlters, IEE Proc. Circuits, Devices, Syst., submitted. [8] J.N. Hallberg, C.S. Lindquist, Nomographs and ﬁlters, J. Franklin Inst. 302 (1976) 145. [9] C.S. Lindquist, C.A. Corral, On the construction of transitional ﬁlter nomographs, J. Franklin Inst., submitted. [10] J.P. Bobis, ‘Energy’ of low pass ﬁlters, Proc. IEEE 51 (1963) 481. [11] E.F. Vandivere, Noise output of a multipole ﬁlter relative to that of the ideal square-response ﬁlter, Proc. IEEE 51 (1963) 1771. [12] D.T. Comer, VCO jitter reduction with bandpass ﬁltering, Electron. Lett. 31 (3) (1995) 11–12. [13] D.-Y. Yu, S.-Q. Wu, L.-M. Li, Design of partial response data transmission ﬁlters with speciﬁed stopband attenuation, IEE Proc.-G 138 (3) (1991) 347–350.

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