- Email: [email protected]
Selectivity nomographs for classical filters
Journal of the Franklin Institute 339 (2002) 61–76
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
62
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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].
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
63
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
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
64
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
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
65
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
66
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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).
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
67
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.
68
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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.
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
69
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
70
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
71
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.
72
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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.
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
73
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.
74
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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.
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
75
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.
76
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
[14] C.A. Corral, C.S. Lindquist, P.B. Aronhime, Sensitivity of the band-edge selectivity of various classical filters, Proceedings of the 40th Midwest Symposium on Circuits Systems, Sacramento, California, Vol. 1, 1997, pp. 324–327. [15] M.D. Lutovac, D.M. Rabrenovic, Improved version of the inverse Chebyshev filter, Electron. Lett. 26 (16) (1990) 1256–1257. [16] K. H!ajek, J. Sedl!acek, A new TICFU transitional approximation, Proc. ECCTD Istanbul 1 (1995) 913–916. [17] C.A. Corral, C.S. Lindquist, Selectivity of elliptic filters, IEE Proc. Circuits, Devices Syst. 147 (3) (2000) 188–195. [18] A. Barua, Novel enhanced pole selectivity bandpass filter with large dynamic range, IEEE Asia Pacific Conference on Circuits Systems, Seoul, South Korea, 1996, pp. 57–60. [19] G. van Ruymbeke, F. Krummenacher, A programmable continuous-time filter, IEEE Int. Symp. Circuits Syst. 2 (1993) 1263–1266. [20] W.-K. Chen (Ed.), The Circuits and Filters Handbook, CRC Press, Salem, MA, 1995.
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
62
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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].
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
63
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
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
64
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
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
65
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
66
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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).
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
67
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.
68
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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.
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
69
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
70
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
71
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.
72
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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.
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
73
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.
74
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
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.
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
75
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.
76
C.S. Lindquist, C.A. Corral / Journal of the Franklin Institute 339 (2002) 61–76
[14] C.A. Corral, C.S. Lindquist, P.B. Aronhime, Sensitivity of the band-edge selectivity of various classical filters, Proceedings of the 40th Midwest Symposium on Circuits Systems, Sacramento, California, Vol. 1, 1997, pp. 324–327. [15] M.D. Lutovac, D.M. Rabrenovic, Improved version of the inverse Chebyshev filter, Electron. Lett. 26 (16) (1990) 1256–1257. [16] K. H!ajek, J. Sedl!acek, A new TICFU transitional approximation, Proc. ECCTD Istanbul 1 (1995) 913–916. [17] C.A. Corral, C.S. Lindquist, Selectivity of elliptic filters, IEE Proc. Circuits, Devices Syst. 147 (3) (2000) 188–195. [18] A. Barua, Novel enhanced pole selectivity bandpass filter with large dynamic range, IEEE Asia Pacific Conference on Circuits Systems, Seoul, South Korea, 1996, pp. 57–60. [19] G. van Ruymbeke, F. Krummenacher, A programmable continuous-time filter, IEEE Int. Symp. Circuits Syst. 2 (1993) 1263–1266. [20] W.-K. Chen (Ed.), The Circuits and Filters Handbook, CRC Press, Salem, MA, 1995.