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On the construction of transitional filter nomographs
Journal of the Franklin Institute 339 (2002) 77–102
On the construction of transitional filter nomographs Claude S. Lindquista,*,1, Celestino A. Corralb b
a University of Miami, P.O. Box 24-8294, Miami, FL 33124, USA Motorola, Inc. 1500 Gateway Blvd., M/S 71, Boynton Beach, FL, 33467, USA
Abstract The construction of nomographs for transitional classical filters is described. Gain functions of classical filters are related to filter requirements resulting in a formulation for the general gain nomograph. The transitional filters that are products of approximating polynomials are incorporated into the general gain nomograph resulting in transitional filter nomographs that are sums of the individual nomographs. Nomographs for transitional filters using alternative forms where poles are interpolated are also considered. The resulting nomographs allow for quick optimization of transitional filter frequency response in many cases. Design examples are submitted and discussed. The proposed transitional filter nomographs provide the engineer with increased insight into the selection of classical transitional filters with optimum frequency response. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved.
1. Introduction NomographsFand their closely related kin, slide rulesFwere once essential tools in engineering design. Filter nomographs, in particular, captured data and information in a useful format, becoming marvelous design aids. In the age of computers, however, these powerful tools have been relegated to museums. Filter design applications or programmed calculators have apparently eliminated the need for this powerful graphical tool. This is an unfortunate development because filter nomographs allow the engineer to evaluate filter performance capability at a glance. *Corresponding author. Tel.: +1-831-457-9206. E-mail address: [email protected] (C.S. Lindquist). 1 Mailing address: 686 Paradise Park, Santa Cruz, CA 95060. 0016-0032/02/$ 22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 2 ) 0 0 0 1 6 - 9
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Nomographs can offer greater insight than formulae or computer algorithms. In this paper we will describe transitional filters and their nomographs. Filter nomographs have proven to be an effective design tool in determining filter order [1–5] and design tradeoffs for frequency response optimization [6]. They present equations in graphical form and thereby provide pictorial insight and greater understanding in the design process. Design nomographs are available for a variety of classical filters including all-pole Butterworth and Chebyshev filters as well as elliptic and inverse Chebyshev filters incorporating transmission zeros [4,5]. However, nomographs have not been made available for transitional filters. This paper describes the construction of transitional filter nomographs from classical filter nomographs. A transitional filter is any filter that results from the combination of the frequency characteristics of any two filters. In general, one of the filters is a constant magnitude type and the other is a constant delay type. Such transitional filters are anticipated to exhibit good magnitude and delay characteristics [4]. Some of the better-known transitional filters include Butterworth–Thomson (TBT) [7], Legendre–Thomson (TLT) [8], Gaussian to 6 or 12 dB [3], Chebyshev-equiripple delay (TCERD) [9], Papoulis-equiripple delay (TPERD) [9], Butterworth–Chebyshev (TBC) [10,11] and transitional ultraspherical–ultraspherical [12]. There are other filters that exhibit transitional filter characteristics but are derived using classical curves like parabolic [13], catenary [14], elliptic contours [15], and circles of Cassini for filters that are maximally flat magnitude beyond the origin (MFMBO) [16,17]. Yet other transitional filters seek to combine the poles of respective responses to achieve a desired response [18]. Complementary pole-pair filters allow for arbitrary pole distribution on a circle akin to a Butterworth filter but with defined angles [19]. All-pole transitional Bessel–Chebyshev filters allow for near equiripple passband magnitude response with near flat delay [20]. The transitional inverse Chebyshev Feistel-Unbehauen (TICFU) filter exploits zeros of transmission for greater selectivity while maintaining good delay characteristics [21]. Other approaches exploit maximally flat magnitude with ripple response to achieve lumped element realizations without transformers [22,23]. Overall, the ability of providing a single implementation compromising magnitude and delay responses make transitional filters ideal in many different applications. This paper presents the construction of nomographs for transitional filters based on classical filter nomographs as advanced in [5]. We begin by reviewing the general gain nomographs which show the relation between maximum passband ripple Mp ; minimum stopband rejection Ms ; frequency o; filter polynomial Fn ðoÞ; and order n: We will also discuss some of the classical filter polynomials. Next, we shall consider transitional filters that utilize these polynomials. One form of transitional filters using two classical polynomials have nomograph curves that are the sum of the two classical filter nomograph curves. Alternative transitions implementing root interpolation are also discussed: The nomograph is constructed directly from these roots and not from the sum of nomographs. Finally, design examples will be presented to demonstrate the performance capabilities of transitional filters.
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2. Gain functions of classical filters Many classical filter gains have the general form 1 ; jHðjoÞj2 ¼ 1 þ e2 Fn2 ðoÞ
ð1Þ
where the filter generating polynomial Fn ðoÞ of order n satisfies Fn ð1Þ ¼ 1 and 0pFn2 ðoÞp1 for jojp1: The form of Eq. (1) allows a quick association between magnitude response and approximation polynomial. It can be used to relate the polynomial to tabulated pole results obtained iteratively, as will be shown. The gain polynomials of several standard classical filters are listed in Table 1. Now we determine the relation between the maximum passband ripple Mp ; minimum stopband rejection Ms ; and filter polynomial Fn of order n: 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 ð2Þ ¼ Mp2 1 þ e2 Fn2 ð1Þ so solving for Mp gives Mp2 ¼ 1 þ e2 :
ð3Þ
The standard gain equation (1) can therefore be expressed as 1 jHðjoÞj2 ¼ : 1 þ ðMp2 1ÞFn2 ðoÞ
ð4Þ
The minimum stopband attenuation factor equals Ms at some frequency o; so from Eq. (4) 1 1 : ð5Þ ¼ Ms2 1 þ ðMp2 1ÞFn2 ðoÞ We can rearrange Eq. (5) as Ms2 1 ¼ ðMp2 1ÞFn2 ðoÞ:
ð6Þ
Table 1 Gain polynomials of some classical filters Approximation
Gain polynomial 2=n
Sync-tuned Bessel Butterworth MFMBO (n even) MFMBO (n odd) Chebyshev Ultraspherical
½1 þ ðMp
1Þo2 n 1
2=n Mp
1
on Bn ð1=oÞ on o2 b2 n 1 b2 ðo2 b2 Þn þ b2n ð1 b2 Þn þ b2n coshðn cosh1 ðoÞÞ Fna ðoÞ
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The filter order n must therefore satisfy Fn2 ðoÞ
M2 1 ¼ s2 Mp 1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ms2 1 or Fn ðoÞ ¼ Mp2 1
ð7Þ
which is one equation in four variables ðMp ; Ms ; o; nÞ for any filter polynomial F : Therefore, there are 4 1 ¼ 3-degrees-of-freedom in Eq. (7). Defining any three variables determines the fourth. This allows a virtually infinite number of parameter combinations to be used. This complicates the design process but gives enormous flexibility in adjusting filter specifications. We see in Table 1 that the filter polynomials Fn are nonlinear functions of n: Therefore, Eq. (7) is usually nonlinear and sometimes difficult if not impossible to solve in closed form. But such nonlinearities are simple to accommodate and solve using nomographs. Let us consider several different standard filters and determine their orders. From Table 1, the Butterworth filter polynomial equals on so Eq. (7) gives o2n ¼
Ms2 1 : Mp2 1
Solving for the Butterworth filter order n yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log ðMs2 1Þ=ðMp2 1Þ n¼ : log o
ð8Þ
ð9Þ
Since n is seldom an integer, we round up Eq. (9) to the next highest integer. The Chebyshev filter polynomial equals Tn ¼ coshðn cosh1 oÞ from Table 1. Using Tn as the left-hand side of the square root of Eq. (7) gives sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ms2 1 coshðn cosh1 oÞ ¼ : ð10Þ Mp2 1 Solving for the Chebyshev filter order n results in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh1 ðMs2 1Þ=ðMp2 1Þ : ð11Þ n¼ cosh1 o These Butterworth and Chebyshev examples show that when closed-form solutions are available, their functional form is too complicated to visualize parameter interactions. Some filters use tabulated polynomials Pn ðoÞ that have been determined iteratively. Others use tabulated roots from which the polynomials may be computed. For example the catenary, parabolic, and elliptic contour filters fall in this category. Such gain functions can be expressed as 1 1 ¼ ; ð12Þ jHðoÞj2 ¼ Pn ðo=o0 ÞPnn ðo=o0 Þ 1 þ ðMp2 1ÞFn2 ðoÞ Q where Pn ðoÞPnn ðoÞ ¼ ni¼1 ðpi pni þ ðpi þ pni Þo þ o2 Þ; pi are the filter poles, and o0 is the normalized frequency for setting the passband as we shall see. The polynomials of Eq. (12) must be frequency scaled so that Pn ð1=o0 ÞPnn ð1=o0 Þ ¼ Mp2 : Solving
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Eq. (12) for the filter polynomial yields Fn2 ðoÞ ¼
Pn ðo=o0 ÞPnn ðo=o0 Þ 1 Mp2 1
ð13Þ
but it usually has no recognizable or well-known form. Nevertheless, Fn can be used for the nomograph construction. An example of this type of filter is the synchronously tunedq(sync-tuned) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi filter 2=n
which has Pn ¼ ½1 þ ðo=o0 Þ2 n=2 : All of its poles lie at o0 ¼ 1= Mp 1 so that Pn ð1Þ ¼ Mp : Therefore, equating Eqs. (13) and (7), the filter order must satisfy ½1 þ ðMp2=n 1Þo2 n 1 ¼ Ms2 1:
Rearranging Eq. (14) and solving for o gives vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2=n uM s 1 : o ¼ t 2=n Mp 1
ð14Þ
ð15Þ
There is no closed-form solution for n but it must be iteratively solved from Eq. (15).
3. General gain nomograph The general gain nomograph is constructed from Eq. (7) after it is converted into a linear equation by taking the log of both sides as [5] logðMs2 1Þ logðMp2 1Þ ¼ log Fn2 ðoÞ:
ð16Þ
By convention, a parameter g is defined as gn ðoÞ ¼ log Fn2 ðoÞ:
ð17Þ
Eq. (16) is a four-parameter equation involving ðMp ; Ms ; o; nÞ: The nomograph consists of two vertical axes labeled Mp and Ms ; and a graph of g versus o for various n: The Mp -axis and Ms -axis are the same for all filters and related to the gaxis through the basic determinant. What differs are the gn ðoÞ graphs. The axis construction of nomographs has been well described [5,6] and the nomograph has the structure shown in Fig. 1. All that is needed is to define a new filter and nomograph with Fn ðoÞ and to plot the resulting gn ðoÞ using Eq. (17). What is noteworthy about the resulting nomographs is the ease with which it is possible to relate all variables. The procedure for determining filter order begins by first expressing the required frequency response in terms of the pertinent parameters: (a) Mp ¼ the maximum attenuation (dB) in the passband. (b) Ms ¼ the maximum attenuation (dB) in the stopband. (c) Os ¼ fs =fp ¼ the normalized stopband frequency. The value of Mp and Ms are entered on the corresponding axes. A line is drawn from the point on the Mp curve through the Ms point until it strikes the g-axis. A horizontal line is then drawn across the graph corresponding to the constant g value.
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Fig. 1. (a) Basic gain nomograph showing axes and supports. (b) Determining filter order from frequency requirements.
A vertical line is drawn through the normalized stopband frequency Os : The horizontal and vertical lines intersect at some point in the graph. The minimum required order is given by the first curve lying above that point as shown in Fig. 1b. Since the nomograph effectively captures all frequency requirements in a graphical form it is easy to trade off parameters to achieve some desired characteristic. This has been treated successfully in [6] and will be expanded in Section 9 for transitional filters. It will become evident that some classical filters display transitional characteristics of their own. However, when the optimization technique is implemented for transitional filters, the relative characteristics of each filter is emphasized and then combined. Because the nomographs used here are general in form, they can be used to optimize transitional filter responses.
4. Transitional filters We have already discussed some of the classical filters and their Fn in Table 1. Now we consider transitional filters based on two classical filters having polynomials F1ðnkÞ of order ðn kÞ and F2k of order k; respectively. The transitional filter gain is
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defined as jHðoÞj2 ¼
1 1þ
e2 F12ðnkÞ ðoÞF22k ðoÞ
:
ð18Þ
The polynomial product has order n so the order of the transitional filter is maintained. The k=n ratio controls the transition between the two limiting filters where 0pk=np1: When k=n ¼ 0; the filter is an F1 type; when k=n ¼ 1; the filter is an F2 type. Setting k=nD1=2 gives the most intermediate filter. We anticipate that this transitional filter will in some manner combine the complex gain responses (i.e. magnitude jHj; phase arg H; delay d arg H=do) of the two filters. For example, the maximally flat magnitude response of the Butterworth filter combined with the equiripple magnitude response of the Chebyshev filter should produce a transitional filter with less inband gain ripple and less delay peaking. Another example is to combine the maximally flat-magnitude Butterworth filter with the maximally flat-delay response Bessel filter. We expect the transitional filter to have a less selective magnitude response and less peaking in the delay response. Several such combinations have been proposed in the past and discussed. Our purpose now is to describe the nomographs of transitional filters and their construction, pointing out the possible combinations of responses. Applying Eqs. (1) and (17) analogy to the transitional gain of Eq. (18), then the nomograph parameter equals gn ðoÞ ¼ log½F12ðnkÞ ðoÞF22k ðoÞ ¼ log F12ðnkÞ ðoÞ þ log F22k ðoÞ ¼ g1ðnkÞ ðoÞ þ g2k ðoÞ:
ð19Þ
This is a most simple but powerful result. Any nth-order transitional filter having the form of Eq. (18) is made up of two filters of order ðn kÞ and k; respectively. In terms of nomographs, its gn curve is simply the sum of its two composite g1ðnkÞ and g2k curves. These two individual nomograph curves are added together to produce the nomograph curve for the transitional filter. The transition methods used in Eq. (18) results in the simplest nomograph construction. For constant n; then k can be chosen to equal any value in the range 0pkpn: Therefore, there are ðn þ 1Þ pairs to use in Eq. (19). These pairs are: ðn; 0Þ; ðn 1; 1Þ; y; ð1; n 1Þ; ð0; nÞ: These pairs result in ðn þ 1Þ different transitional filters of fixed order n: These can be plotted on the same nomograph to provide a ready reference for transitional filters.
5. Design examples Let us now demonstrate the nomograph construction with several examples. We begin by considering the transitional Butterworth–Chebyshev filter, with gain jHðoÞj2 ¼
1 1þ
e2 o2ðnkÞ Tk2 ðoÞ
:
ð20Þ
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The g parameter therefore equals gn ðoÞ ¼ logðo2ðnkÞ Þ þ logðcoshðk cosh1 ðoÞÞ2 Þ:
ð21Þ
Let us consider the case where order n ¼ 5: The six different order combinations ðn k; kÞ are ð5; 0Þ; ð4; 1Þ; ð3; 2Þ; ð2; 3Þ; ð1; 4Þ; and ð0; 5Þ: The individual Butterworth and Chebyshev nomographs are shown in Fig. 2. If we choose n ¼ 0–5 curves and pair them as above, we can employ Eq. (21) and plot the resulting nomograph as shown in Fig. 3. The six different TBC filters have the same asymptotic slope (d log jHj=d log o ¼ 5) for o-10 but different curvatures from the nomograph origin at o ¼ 1: These curvatures are more than Butterworth but less than Chebyshev except at the extremes. This example shows the combination of a high-selectivity high-delay distortion Chebyshev filter with a moderate-selectivity moderate-delay Butterworth filter. The result is a compromise filter that has higher selectivity than the Butterworth filter but lower delay distortion than the Chebyshev filter. 2=n Another example is the synchronously tuned filter which has Pn ¼ ð1 þ ½Mp 2 n=2 1o Þ : Normally we would use a Bessel filter but we cannot write a closed-form expression for its gain. The sync-tuned and Bessel filters have fairly similar magnitude and delay characteristics, so we can substitute the sync-tuned to demonstrate the process. Using the Butterworth polynomial, the g parameter therefore equals gn ðoÞ ¼ logðo2ðnkÞ Þ þ logð½1 þ ½Mp2=k 1o2 k Þ:
ð22Þ
This TBS nomograph is shown in Fig. 4. Here we have combined two nomographs, one with straight lines (Butterworth) and one concave downward (sync-tuned). This combines a moderate-selectivity moderate-delay-distortion Butterworth filter with a low-selectivity low-delay-distortion sync-tuned filter. The produces a lower selectivity filter with lower delay distortion. Still another example is the Chebyshev–sync-tuned transitional filter. Its characteristics will be very close to a Chebyshev–Bessel transitional filter. The g parameter for this combination is gn ðoÞ ¼ logðcoshððn kÞ cosh1 ðoÞ2 Þ þ logð½1 þ ½Mp2=k 1o2 Þk Þ:
ð23Þ
This transitional Chebyshev–sync-tuned (TCS) nomograph is shown in Fig. 5. The resulting nomograph shows the combination of one curve that is concave upward (Chebyshev) with another that is concave downward (sync-tuned). Therefore, we are combining a high-selectivity high-delay-distortion Chebyshev filter with a lowselectivity low-delay-distortion sync-tuned filter. Evidently, we can obtain an intermediate filter that has a nomograph close to that of a Butterworth filter (cf. k ¼ 3 or 4 with Fig. 2b). We can also present some recent examples of nontransitional filters that can be converted into transitional filters. A Butterworth-inverse–Chebyshev transitional filter [23] would have gn ðoÞ ¼ logðo2ðnkÞ Þ þ logðcoshðk cosh1 ðos =oÞÞ2 Þ;
ð24Þ
where os is the normalized stopband frequency and the inverse Chebyshev formulation is from [24]. A MFMBO–Chebyshev transitional filter
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85
Fig. 2. Gain nomographs for (a) sync-tuned, (b) Butterworth, and (c) Chebyshev.
would have gn ðoÞ ¼ log
2 nk o b2 þlogðcoshðk cosh1 ðoÞÞ2 Þ 1 b2
ð25Þ
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Fig. 3. Nomograph of fifth-order transitional Butterworth–Chebyshev filter.
for n even and gn ðoÞ ¼ log
b2ðnkÞ þ ðo2 b2 Þnk b2ðnkÞ þ ð1 b2 Þnk
! þ logðcoshðk cosh1 ðoÞÞ2 Þ
ð26Þ
for n odd. The parameter b is selected for providing maximally flat magnitude slope condition at frequency 0pbp1 and is a shaping control. We note in passing that any number m of filters can be transitionally combined as jHðoÞj2 ¼
1þe
1 Q 2
m
Fm2 i
;
X
mi ¼ n
in which case the nomograph parameter equals X X gðoÞ ¼ logðFm2 i Þ mi ¼ n:
ð27Þ
ð28Þ
m
Here the transitional nomograph is always the average of m individual nomographs.
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Fig. 4. Nomograph of fifth-order transitional Butterworth–sync-tuned filter.
6. Transitional filters having alternative forms The second standard form for transitional filter gain involves interpolating the poles and zeros of two filters, denoted as s1 and s2 ; respectively. One interpolation is Form A :
jsi j ¼ js1i jm js2i j1m ;
yi ¼ my1i þ ð1 mÞy2i ;
ð29Þ
where y1i ¼ argðs1i Þ and y2i ¼ argðs2i Þ: The root magnitudes are geometrically interpolated and the root angles are arithmetically interpolated. The transition parameter 0pmp1: Taking the logarithm of the magnitude shows log jsi j ¼ m log js1i j þ ð1 mÞ log js2i j
ð30Þ
and that the magnitude interpolation is geometric. Another magnitude interpolation that can be employed is Form B :
jsi j ¼ mjs1i j þ ð1 mÞjs2i j;
yi ¼ my1i þ ð1 mÞy2i :
ð31Þ
In this case, the root magnitudes are arithmetically interpolated like the phases. Other interpolation schemes like geometrically interpolated angle can be considered (although they have not been proposed in the past) but all the results are processed as follows:
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Fig. 5. Nomograph of fifth-order transitional Chebyshev–sync-tuned filter.
The transitional filter transfer function HðsÞ is then k k ¼ Qn : HðsÞ ¼ Qn jyi i¼1 ðs þ si Þ i¼1 ðs þ jsi je Þ In ac steady state, s ¼ jo and the squared gain equals k jHðoÞj2 ¼ Qn : 2 2 ðjs j þ 2ojs i i j sinðyi Þ þ o Þ i¼1
ð32Þ
ð33Þ
Therefore, this gain takes the polynomial form of Eq. (12). The gain polynomial must then be determined using Eq. (13) which yields n Y Fn2 ðoÞ ¼ ðjsi j2 þ 2ojsi j sinðyi Þ þ o2 Þ 1: ð34Þ i¼1
Using the geometric interpolation form of Eq. (29), this gain polynomial becomes n Y Fn2 ðoÞ ¼ ðjs1i j2m js2i j2ð1mÞ þ 2ojs1i jm js2i j1m i¼1
sinðmy1i þ ð1 mÞy2i Þ þ o2 Þ 1:
ð35Þ
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Using the other arithmetic interpolation form of Eq. (31), this gain polynomial becomes n Y ðm2 js1i j2 þ ð1 mÞ2 js2i j2 þ 2mð1 mÞjs1i j js2i j Fn2 ðoÞ ¼ i¼1
þ 2oðmjs1i j þ ð1 mÞjs2i jÞ sinðmy1i þ ð1 mÞy2i Þ þ o2 Þ 1:
ð36Þ
Unfortunately, these polynomial forms are usually not recognizable functions as in the case of Eq. (1). Therefore, the gain nomographs are plotted directly using Eqs. (35) and (36), or some other resulting interpolation formula. These interpolations do not result in nomographs that are sums of constituent nomographs. Another comment should be made. The roots used in the interpolations of Eq. (29) or Eq. (31) are usually normalized so the two respective filters have unit passband bandwidth. When the root magnitudes are interpolated, the resulting transitional filter generally does not have unit bandwidth but bandwidth o0 : Therefore, the root magnitudes must be reduced or increased by the frequency scaling factor o0 as m 1m s1 i s2 i s1i s2 ð37Þ jsi j ¼ or jsi j ¼ m þ ð1 mÞ i o0 o0 o0 o0 before constructing the gain nomograph using Eq. (35) or Eq. (36). The root angles are unchanged. To demonstrate these interpolation methods, reconsider the Butterworth–Chebyshev fifth-order transitional filter discussed in Eq. (21). The Butterworth poles are s1 ¼ 1:0000;
js1 j ¼ 1:0000;
y1 ¼ 01;
s2 ; s3 ¼ 0:80907j0:5878;
js2 j; js3 j ¼ 1:0000;
y2 ; y3 ¼ 7361;
s4 ; s5 ¼ 0:30907j0:9511;
js4 j; js5 j ¼ 1:0000;
y4 ; y5 ¼ 7721:
ð38Þ
These angles are with respect to the negative real axis. The Butterworth poles lie on a unit circle with angles that are multiples of 1801=5 ¼ 361: The Chebyshev poles are s1 ¼ 0:1775;
js1 j ¼ 0:1775;
y1 ¼ 01;
s2 ; s3 ¼ 0:14357j0:5970;
js2 j; js3 j ¼ 0:6140;
y2 ; y3 ¼ 776:51;
s4 ; s5 ¼ 0:05497j0:9659;
js4 j; js5 j ¼ 0:9762;
y4 ; y5 ¼ 786:71:
ð39Þ
The poles lie on an ellipse having a semi-minor axis length of 0.1775 and a semimajor axis length of one. Both sets of poles given by Eqs. (38) and (39) have been adjusted to produce 3 dB passband ripple and unit ripple bandwidth. Since o0 ¼ 1; no frequency scaling of roots is needed as discussed in Eq. (37). Consider the geometric interpolation of Eq. (29). The TBC filter poles are js1 j ¼ 1m 0:17751m ;
y1 ¼ 01;
m
1m
;
y2 ; y3 ¼ 7½m361 þ ð1 mÞ76:51;
m
1m
;
y4 ; y5 ¼ 7½m721 þ ð1 mÞ86:71:
js2 j; js3 j ¼ 1 0:6140 js4 j; js5 j ¼ 1 0:9762
ð40Þ
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The pole locations move from the Chebyshev ellipse (m ¼ 0) to the Butterworth circle (m ¼ 1) as m is increased from zero to unity as shown in Fig. 6a. The five root loci are almost horizontal so the imaginary part of the poles are almost constant. The resulting gain nomograph is shown in Fig. 6b. The intermediate filters occur for mD1=2: Comparing the TBC filter given by Eq. (21) whose nomograph is shown in Fig. 3, we see little difference. If we label the curves of Fig. 3 as k=n ¼ ð0; :1; :2; y; 1Þ; then mDk=n in Fig. 6b. Now consider the arithmetic interpolation of Eq. (31). The TBC filter now has js1 j ¼ m þ ð1 mÞ0:1775;
y1 ¼ 01; y2 ; y3 ¼ 7½m361 þ ð1 mÞ76:51; y4 ; y5 ¼ 7½m721 þ ð1 mÞ86:71:
js2 j; js3 j ¼ m þ ð1 mÞ0:6140; js4 j; js5 j ¼ m þ ð1 mÞ0:9762;
ð41Þ
The pole angles of Eq. (41) are the same as those used in Eq. (40). The root locus is shown in Fig. 7a. The pole vector lengths shorten by equal amounts while the angles also decrease by equal amounts. Comparing this root locus with that of the other interpolation shown in Fig. 6a, we see that path is more curved. This slight difference is minor in the magnitude responses, more noticeable in the phase responses, and particularly noticeable in the delay responses. There is negligible difference in the impulse and step time domain responses. The nomograph is shown in Fig. 7b. Again mD1=2 gives an intermediate response. There is little difference between Figs. 6 and 7 so mDk=n: To show more dramatic differences, reconsider the Chebyshev–sync-tuned fifthorder transitional filter discussed in Eq. (22). The sync-tuned poles are s1 ; s2 ; s3 ; s4 ; s5 ¼ 2:593;
js1 j; js2 j; js3 j; js4 j; js5 j ¼ 2:593;
y1 ; y2 ; y3 ; y4 ; y5 ¼ 01:
ð42Þ
All poles are located at 2:593: The 3 dB filter bandwidth is unity. The Chebyshev poles were given in Eq. (39) and also have unit 3 dB bandwidth. Using the geometric interpolation of Eq. (29), the TCS filter poles are js1 j ¼ 2:593m 0:17751m ;
y1 ¼ 01;
m
1m
;
y2 ; y3 ¼ 7½m01 þ ð1 mÞ76:51;
m
1m
;
y4 ; y5 ¼ 7½m01 þ ð1 mÞ86:71:
js2 j; js3 j ¼ 2:593 0:6140 js4 j; js5 j ¼ 2:593 0:9762
ð43Þ
The root locus is shown in Fig. 8a and the nomograph in Fig. 8b. The arithmetic interpolation of Eq. (31) gives a TCS filter with poles js1 j ¼ m2:593 þ ð1 mÞ0:1775;
y1 ¼ 01;
js2 j; js3 j ¼ m2:593 þ ð1 mÞ0:6140;
y2 ; y3 ¼ 7½m01 þ ð1 mÞ76:51;
js4 j; js5 j ¼ m2:593 þ ð1 mÞ0:9762;
y4 ; y5 ¼ 7½m01 þ ð1 mÞ86:71:
ð44Þ
The root locus is shown in Fig. 9a and the nomograph in Fig. 9b. The geometric interpolation gives a flatter root loci than the arithmetic interpolation. But both loci have considerably more arc than the earlier TBC filter. Both nomographs have more curvature which reflects the root loci arcs. The TCS filter plots of Fig. 5 using
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Fig. 6. Fifth order transitional Butterworth–Chebyshev filter with geometric interpolated poles. (a) Root locus and (b) nomograph.
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Fig. 7. Fifth order transitional Butterworth–Chebyshev filter with arithmetic interpolated poles. (a) Root locus and (b) nomograph.
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Fig. 8. Fifth order transitional Chebyshev–sync-tuned filter with geometric interpolated poles. (a) Root locus and (b) nomograph.
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Fig. 9. Fifth order transitional Chebyshev–sync-tuned filter with arithmetic interpolated poles. (a) Root locus and (b) nomograph.
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polynomial interpolation rather than root interpolation are fairly similar so mDk=n: However the root locus spans a much larger area and is largely circular unlike the TBC filters of Figs. 6 and 7. 8. Transitional filters using transmission zeros We now consider transitional filter nomographs that arise using k pairs of transmission zeros. The gains of such filters often take the form 1 jHðoÞj2 ¼ ; k=2on; ð45Þ Qk=2 2 2 2 1 þ e Fn ðoÞ=c i¼1 ðo o2i Þ2 Q 2 where Fn is a filter polynomial of order n and ðo o2i Þ is the transmission zero polynomial of order k: For the filter to remain lowpass and have jHðNÞj ¼ 0; k must be chosen so ko2n: The nomographs of such equations are simple to construct numerically but are extremely difficult to evaluate analytically. Let us demonstrate this using a Butterworth filter having Fn2 ¼ o2n and one pair of transmission zeros 7o0 : Then the gain equals 1 : ð46Þ jHðoÞj2 ¼ 1 þ ð1 o20 Þ2 o2n =ðo2 o20 Þ2 The transitional filter order is n and the asymptotic rolloff rate is 20ðn 2Þ dB=dec: The dc gain jHð0Þj is unity and the 3 dB corner frequency has been normalized to unity since jHð1Þj2 ¼ 1=2 by the use of the ð1 o20 Þ2 factor. If o0 ¼ N; the filter is Butterworth. To obtain greater band-edge selectivity, the transmission zero o0 is chosen as 1oo0 oN to obtain the desired stopband rejection. The case when o0 ¼ 2 is shown in Fig. 10 for n ¼ 2–5. We see that the gain rapidly approaches zero beyond the 3 dB band-edge frequency of unity. The gain equals zero at o0 ¼ 2 and then rebounds to some value Ms beyond o0 : For plotting nomographs, only the gain portion from o ¼ 1 to os where jHðos Þj ¼ 1=Ms is used. Then Ms and os must be determined. This is done as follows: The minimum stopband rejection Ms occurs at the stopband peaking frequency opeak where djHj2 =do2 ¼ 0: Solving Eq. (46) leads to the simple result that opeak ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o0 n=ðn 2Þ: The minimum stopband rejection Ms is obtained by evaluating jHðopeak Þj ¼ 1=Ms to obtain n n n 2 2 n1 2 Ms Do0 ðo0 1Þ : ð47Þ n2 2 Then using Eq. (46), the stopband frequency os satisfies n n n 2 2 on ðo2 1Þ 2 Ms D s 2 0 2 Don1 ðo 1Þ 0 0 n2 2 ðos o0 Þ
ð48Þ
which is highly nonlinear and must be solved numerically for os : To determine the nomograph, we must find the op and Ms values for o > o0 : We then extend Ms backwards until it intersects the gain characteristic for 1oooo0 : We then plot the gðoÞ in this range. Using this technique, the Butterworth-single
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Fig. 10. Magnitude response of Butterworth filter for n ¼ 2; 3; 4; 5 and zero at o0 ¼ 2:
transmission zero nomograph is shown in Fig. 11 for o0 ¼ 2: The unused portion of the filter gain characteristic is shown by solid curves. Another useful combination of single transmission zero nomographs are shown in Fig. 12. This shows the cases for o0 ¼ 2; 3; y; 9: We see the Butterworth nomograph curves which are the straight line segments that connect the ð0; 1Þ origin to gn ¼ 4; 6; y; 20 at o ¼ 10 for n ¼ 2; 3; y; 10; respectively. The single transmission zero produces more stopband attenuation. This produces the tangentially upward curves shown in Fig. 12. The nomograph parameters gn ðoÞ equals
2 o o20 2n gn ðoÞ ¼ logðo Þ log ; n>2 ð49Þ 1 o20 by viewing Eq. (46) as having the form of Eq. (18) and the parameters of Eq. (19). Notice the negative sign of the transmission zero term in this formulation. Asymptotically, the Butterworth gain slope is 20n dB=dec while the transmission zero gain slope is 40 dB=dec: This emphasizes that transmission zeros improve (i.e. increase) the band-edge selectivity but reduce the asymptotic attenuation by 40 dB=dec: Multiple zeros can be used but they must generally be aligned by some analytical technique as with inverse Chebyshev and elliptic filters, or by some numerical technique like the Remez algorithm or alternatives [25]. Regardless of what approach is used, the resulting transmission zero function can be used to form the nomograph as described above. The nomograph parameter equals 0 1 Y 2 gn ðoÞ ¼ logðFnþk Þ log@c ðo2 o20 ÞA; kon=2 ð50Þ k=2
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Fig. 11. Nomograph of Butterworth filter with transmission zero at normalized frequency 2. Nomograph should use right-hand portion beyond inflection point.
for a 40n dB=dec rolloff filter. General gain functions with arbitrary pole and zero ripples require a different approach. Many numerical problems arise when constructing the nomograph of general gain functions. Consider the general attenuation function shown in Fig. 13(a). The attenuation ripples in both the passband (below passband frequency op ) and stopband (above stopband frequency os ). Both bands exhibit non-equiripple response. The maximum passband attenuation is Mp and the minimum stopband attenuation is Ms : These levels and frequencies are shown and labeled in Fig. 13(a). When constructing the nomograph, only the transition gain characteristic between op and os is used; the rest of the characteristic is ignored. The four parameters used in the nomograph are ðMp ; op ; Ms ; os Þ and the nomograph construction of general filters begins by solving for these parameters from four equations using numerical methods. Then Mp and Ms are used to determine g and Os ¼ os =op : The nomograph curves are plotted for fixed filter order n as a function of ðg; Os Þ for 0pgp12 and 1pOs p10:
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Fig. 12. Single transmission zero showing greater stopband attenuation initially at the expense of decreased asymptotic attenuation.
Fig. 13. (a) Definition of filter attenuation jHj2 and (b) mapping to nomograph parameter g:
We now describe the steps which enable us to determine ðMp ; op ; Ms ; os Þ: The steps involve the general filter gain which has the form Q ðjo þ pi Þðjo þ pni Þ jHðjoÞj2 ¼ 1 þ Q n ; ð51Þ n m ðjo þ zi Þðjo þ zi Þ where the denominator represent zeros of transmission when Reðzi Þ ¼ 0: The first step is to find the passband peaking frequencies o1 from djHj2 ¼ 0; 0po1 pop : ð52Þ do2 o¼o1
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In Fig. 13(a), there are five such frequencies (i.e., peaks and valleys). These values are determined by solving Eq. (52) using numerical methods. The second step is to select the maximum-peaking frequency o1 that gives the maximum passband attenuation Mp and to find the corresponding passband frequency op : This is accomplished by solving max jHðjo1 Þj2 ¼ Mp2 ¼ jHðjop Þj2 :
ð53Þ
In Fig. 13(a), the five frequencies are found from Eq. (52). These frequencies are substituted into the gain in Eq. (51). The maximum attenuation value is defined to be Mp and the frequency which produces it is o1 : After Mp is thus determined, numerical methods are used to find the frequency op that satisfies Eq. (53). This concludes the steps necessary for evaluation of the passband. The third step is to find the stopband peaking frequencies o2 where djHj2 ¼ 0; do2 o¼o2
os po2 pN:
ð54Þ
In Fig. 13(a), there are seven such frequencies (i.e., peaks and valleys) which are found by solving Eq. (54) by numerical techniques. The fourth and final step is to select the minimum-peaking frequency o2 that gives the minimum stopband attenuation Ms and to find the corresponding stopband frequency os : This is accomplished by solving min jHðjo2 Þj2 ¼ Ms2 ¼ jHðjos Þj2 :
ð55Þ
In Fig. 13(a), seven frequencies are found from Eq. (54). These frequencies are substituted into the gain in Eq. (51). The minimum attenuation value is defined to be Ms and the frequency which produces it is o2 : After Ms is thus determined, numerical methods are used to find the frequency op that satisfies Eq. (55). This concludes the steps necessary for evaluation of the stopband and results in a nomograph such as shown in Fig. 13(b). The problem with this technique is that the derivatives in Eqs. (52) and (54) are highly nonlinear equations and cannot generally be solved in closed form. The same can be said of frequencies involved in Eqs. (53) and (55). Therefore, numerical methods must be used to determine the parameters used to form the nomograph, but the end result is a graphical rendition of the desired filter characteristics. This is unnecessary for classical filter nomographs. The all-pole filters have no stopband ripple. Therefore, the derivative of step 3 using Eq. (54) is unnecessary and Eq. (55) has a closed-form nonlinear solution. In the passband, the classical filter polynomials have been chosen to be bounded by known ripple (often jFn jp1) over the range 0pop1 and monotonically increases beyond that range. Thus, the first step of finding the derivative in Eq. (52) is unnecessary and Eq. (53) is known by polynomial substitution.
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9. Optimizing transitional filters using nomographs In [6] the authors proposed a technique for optimizing classical filter magnitude response based on filter nomographs as those presented here. The optimization was the achievement of any of the following without increasing filter order: 1. Maximizing band-edge selectivity. 2. Minimizing passband ripple. 3. Maximizing stopband rejection. The method consisted of adjusting the filter requirements to obtain a new set of poles/zeros for the transfer function. The resulting filter used all degrees of freedom (except order) to achieve the desired optimization. The filter nomographs considered here effectively capture all requirements (degrees of freedom) for the design. Since the order must be an integer, any ‘‘excess’’ order is available for optimization. For optimizing the frequency response of a filter as stated above, the values of Mp ; Ms and Os can all be determined readily from the nomograph. In effect, it is possible to move the point in Fig. 1b along a constant g or Os line to the curve corresponding to the required order. This effectively performs the iteration in Eq. (16) for all values depending on Fn : From this optimization technique we can note that several classical filters can be made to have ‘‘transitional’’ characteristics. For example, the elliptic filter can be ‘‘optimized’’ for less delay variation by minimizing passband ripple for the given order. Another option is to reduce passband delay peaking by minimizing selectivity (maximizing transition bandwidth). For Chebyshev filters, passband ripple can be minimized to reduce delay distortion. The degrees of freedom available for optimization can be noted from the selectivity equations as described in [26]. For the case where the transitional filter consists of a combination of two responses, the result is a new Fn that is captured in the nomograph. Two approaches become apparent: (a) Optimize the individual responses prior to combining them in the transitional form. This requires that the engineer know the order of the filter prior to implementation, which can be obtained from the transitional nomograph. (b) Optimize the response directly from the transitional nomograph. This is the most straight-forward way: The requirements are entered in the nomograph and the filter order determined. The optimization described in [6] can then be performed on the transitional nomograph: adjusting Mp ; Ms or Os : In the case of transitional filters implementing pole/zero interpolation, the method is more involved. Here, each response must be treated independently and the corresponding poles and zeros calculated. The interpolation is then implemented to achieve the desired combination. The transitional nomograph for interpolated poles/ zeros can be used to determine the order in most of the cases.
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As an example, let us consider the following filter requirements: Mp ¼ 1:25 dB; Ms ¼ 40 dB; and Os ¼ 3: In either case we could implement a fifth-order transitional Butterworth–sync-tuned or Chebyshev–sync-tuned. For the Butterworth filter, the only degree of freedom is order, so let us choose the Chebyshev–sync-tuned filter for optimization. Entering the passband and stopband requirements into Fig. 5 we get k ¼ 4 for the Chebyshev response. We note that we have some margin because there is excess rejection. We can maximize selectivity by reducing the stopband corner ratio to Os ¼ 2:5: We can also reduce the passband ripple by selecting where the k ¼ 4 curve intersects Os ¼ 3 and reading gE5:3: Drawing a line from this point through Ms ¼ 40 dB gives us a new passband ripple requirement of Mp ¼ 0:2 dB: The advantage of doing this is the reduction of delay ripple variation in the Chebyshev passband, further emphasizing delay response of the transitional filter.
10. Conclusions We have presented the construction of transitional filter nomographs. For the case where two (or more) filter functions are combined, the resulting nomograph is the sum of the individual filter nomographs. This simple yet powerful result means that no additional theoretical work needs to be completed when two potential filters are contemplated to be transitionally combined! Since an nth-order transitional filter can be realized in ðn þ 1Þ different forms using the ðn; 0Þ; ðn 1; 1Þ; y; ð0; nÞ pairs, there are a large varieties of possibilities to be considered. The nomograph can therefore be sketched using the ðn þ 1Þ pairs of curves without resorting to computers, etc. Optimization of these combined responses was also described. For alternative forms using pole and zero interpolation, the nomographs are constructed directly from the interpolation formulas. In certain cases, the resulting transitional filter magnitude response are similar to the combined function case. In general, however, the pole interpolation significantly distorts phase/delay properties, so they merit further investigation. The purpose here was to identify filter transition forms and generate nomographs based on those forms for ease of filter selection and design. With these nomographs and the proposed optimization technique, the engineer can make ready evaluation of the unique capabilities of transitional filters.
References [1] M. Kawakami, Nomographs for Butterworth and Chebyshev filters, IEEE Trans. Circuit Theory CT10 (1963) 288–290. [2] E. Christian, E. Eisenmann, Filter Design Tables and Graphs, Wiley, New York, 1966. [3] A.I. Zverev, Handbook of Filter Synthesis, Wiley, New York, 1967. [4] C.S. Lindquist, Active Network Design with Signal Filtering Applications, Steward & Sons, CA, 1977 (Chapter 4). [5] J.N. Hallberg, C.S. Lindquist, Nomographs and filters, J. Franklin Inst. 302 (1976) 111–145.
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[6] C. Corral, C.S. Lindquist, Design for optimum classical filters, under consideration for IEE Proc. Circuits, Devices and Systems. [7] Y. Peless, T. Murakami, Analysis and synthesis of transitional Butterworth–Thompson filters and bandpass amplifiers, RCA Rev. (1957) 60–94; H.J. Hindin, Impulse response and transfer phase of transitional Butterworth–Thompson filters, IEEE Trans. Circuit Theory CT-15 (1968) 471–474. [8] G.L. Aiello, P.M. Angelo, Transitional Legendre–Thomson filters, IEEE Trans. Circuits Systems CAS-21 (1974) 159–162. [9] A.D. Taylor, A study of transitional filters, M.S.E.E. Directed Research, California State University, Long Beach, August 1975. [10] A. Budak, P. Aronhime, Transitional Butterworth–Chebyshev filters, IEEE Trans. Circuit Theory CT-18 (1971) 413–415. [11] S.C. Dutta Roy, P. Varanasi, Transitional Butterworth–Chebyshev filters, Electron. Lett. 14 (1978) 179–180. [12] J. Altikiouzel, D.T. Phuc, On transitional ultraspherical–ultraspherical filters, Proc. IEEE 66 (1978) 703–706. [13] S.K. Mullick, Pulse networks with parabolic distribution of poles, IRE Trans. Circuit Theory CT-8 (1961) 302–305. [14] M.S. Ghausi, M. Adamowicz, A new class of filters for pulse applications, J. Franklin Inst. 282 (1966) 20–30. [15] E.S. Kuh, Synthesis of lumped parameter precision delay line, Proc. IRE 45 (1957) 1632–1642; J.O. Scanlan, Transfer functions with elliptic distribution of poles at equal frequency spacings, IEEE Trans. Circuit Theory CT-12 (1965) 260–266. [16] P. Aronhime, A. Budak, Maximally flat magnitude beyond the origin, IEEE Trans. Circuit Theory CT-18 (1971) 409–411. [17] C.S. Lindquist, C.A. Corral, Nomographs of MFMBO filters (maximally flat magnitude beyond the origin), Asilomar Conf. Signals, Syst., Computers, 2 (2000) 1635–1638. [18] J.R. Johnson, D.E. Johnson, R.Q. Perritt, R.J. LaCarna, Transitional rational filters, IEEE Trans. Circuits Systems CAS-26 (1979) 976–979. [19] K.R. Pai, K.V.V. Murthy, V. Ramachandran, Complementary pole-pair filtersFA new family of transitional filters, Proc. Int. Symp. Circuits Systems 4 (1993) 2264–2267. [20] P. Aronhime, R. Castagno, All-pole transitional Bessel–Chebyshev filters, Electron. Lett. 25 (1989) 18–19. [21] K. H!ajek, J. Sedl!acek, A new TICFU transitional approximation, Proceedings of the ECCTD Istanbul, Vol. 1, 1995, pp. 913–916. [22] D.Y. Kim, C.H. Yun, S.W. Choi, A modified elliptic function with diminishing ripples, Proceedings of the 37th Midwest Symposium on Circuits Systems, Vol. 2, 1994, pp. 1029–1031. [23] J.-C. Ahn, S.-W. Choi, C.-H. Yun, D.-Y. Kim, Some prominent characteristics of modified inverse Chebyshev function, Proceedings of the 40th Midwest Symposium on Circuits Systems, 1997, pp. 328–331. [24] L. Weinberg, Network Analysis and Synthesis, McGraw-Hill, New York, 1962. # [25] A.C.M. de Queiroz, L.P. Caloba, An approximation algorithm for irregular-ripple filters, Symposium Record, International Telecommunications Symposium (ITS), 1990, pp. 430–433. [26] C.A. Corral, C.S. Lindquist, P.B. Aronhime, Sensitivity of band-edge selectivities, Analog Integrated Circuits and Signal Processing, May 1999, pp. 189–196.
On the construction of transitional filter nomographs Claude S. Lindquista,*,1, Celestino A. Corralb b
a University of Miami, P.O. Box 24-8294, Miami, FL 33124, USA Motorola, Inc. 1500 Gateway Blvd., M/S 71, Boynton Beach, FL, 33467, USA
Abstract The construction of nomographs for transitional classical filters is described. Gain functions of classical filters are related to filter requirements resulting in a formulation for the general gain nomograph. The transitional filters that are products of approximating polynomials are incorporated into the general gain nomograph resulting in transitional filter nomographs that are sums of the individual nomographs. Nomographs for transitional filters using alternative forms where poles are interpolated are also considered. The resulting nomographs allow for quick optimization of transitional filter frequency response in many cases. Design examples are submitted and discussed. The proposed transitional filter nomographs provide the engineer with increased insight into the selection of classical transitional filters with optimum frequency response. r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved.
1. Introduction NomographsFand their closely related kin, slide rulesFwere once essential tools in engineering design. Filter nomographs, in particular, captured data and information in a useful format, becoming marvelous design aids. In the age of computers, however, these powerful tools have been relegated to museums. Filter design applications or programmed calculators have apparently eliminated the need for this powerful graphical tool. This is an unfortunate development because filter nomographs allow the engineer to evaluate filter performance capability at a glance. *Corresponding author. Tel.: +1-831-457-9206. E-mail address: [email protected] (C.S. Lindquist). 1 Mailing address: 686 Paradise Park, Santa Cruz, CA 95060. 0016-0032/02/$ 22.00 r 2002 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 2 ) 0 0 0 1 6 - 9
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Nomographs can offer greater insight than formulae or computer algorithms. In this paper we will describe transitional filters and their nomographs. Filter nomographs have proven to be an effective design tool in determining filter order [1–5] and design tradeoffs for frequency response optimization [6]. They present equations in graphical form and thereby provide pictorial insight and greater understanding in the design process. Design nomographs are available for a variety of classical filters including all-pole Butterworth and Chebyshev filters as well as elliptic and inverse Chebyshev filters incorporating transmission zeros [4,5]. However, nomographs have not been made available for transitional filters. This paper describes the construction of transitional filter nomographs from classical filter nomographs. A transitional filter is any filter that results from the combination of the frequency characteristics of any two filters. In general, one of the filters is a constant magnitude type and the other is a constant delay type. Such transitional filters are anticipated to exhibit good magnitude and delay characteristics [4]. Some of the better-known transitional filters include Butterworth–Thomson (TBT) [7], Legendre–Thomson (TLT) [8], Gaussian to 6 or 12 dB [3], Chebyshev-equiripple delay (TCERD) [9], Papoulis-equiripple delay (TPERD) [9], Butterworth–Chebyshev (TBC) [10,11] and transitional ultraspherical–ultraspherical [12]. There are other filters that exhibit transitional filter characteristics but are derived using classical curves like parabolic [13], catenary [14], elliptic contours [15], and circles of Cassini for filters that are maximally flat magnitude beyond the origin (MFMBO) [16,17]. Yet other transitional filters seek to combine the poles of respective responses to achieve a desired response [18]. Complementary pole-pair filters allow for arbitrary pole distribution on a circle akin to a Butterworth filter but with defined angles [19]. All-pole transitional Bessel–Chebyshev filters allow for near equiripple passband magnitude response with near flat delay [20]. The transitional inverse Chebyshev Feistel-Unbehauen (TICFU) filter exploits zeros of transmission for greater selectivity while maintaining good delay characteristics [21]. Other approaches exploit maximally flat magnitude with ripple response to achieve lumped element realizations without transformers [22,23]. Overall, the ability of providing a single implementation compromising magnitude and delay responses make transitional filters ideal in many different applications. This paper presents the construction of nomographs for transitional filters based on classical filter nomographs as advanced in [5]. We begin by reviewing the general gain nomographs which show the relation between maximum passband ripple Mp ; minimum stopband rejection Ms ; frequency o; filter polynomial Fn ðoÞ; and order n: We will also discuss some of the classical filter polynomials. Next, we shall consider transitional filters that utilize these polynomials. One form of transitional filters using two classical polynomials have nomograph curves that are the sum of the two classical filter nomograph curves. Alternative transitions implementing root interpolation are also discussed: The nomograph is constructed directly from these roots and not from the sum of nomographs. Finally, design examples will be presented to demonstrate the performance capabilities of transitional filters.
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2. Gain functions of classical filters Many classical filter gains have the general form 1 ; jHðjoÞj2 ¼ 1 þ e2 Fn2 ðoÞ
ð1Þ
where the filter generating polynomial Fn ðoÞ of order n satisfies Fn ð1Þ ¼ 1 and 0pFn2 ðoÞp1 for jojp1: The form of Eq. (1) allows a quick association between magnitude response and approximation polynomial. It can be used to relate the polynomial to tabulated pole results obtained iteratively, as will be shown. The gain polynomials of several standard classical filters are listed in Table 1. Now we determine the relation between the maximum passband ripple Mp ; minimum stopband rejection Ms ; and filter polynomial Fn of order n: 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 ð2Þ ¼ Mp2 1 þ e2 Fn2 ð1Þ so solving for Mp gives Mp2 ¼ 1 þ e2 :
ð3Þ
The standard gain equation (1) can therefore be expressed as 1 jHðjoÞj2 ¼ : 1 þ ðMp2 1ÞFn2 ðoÞ
ð4Þ
The minimum stopband attenuation factor equals Ms at some frequency o; so from Eq. (4) 1 1 : ð5Þ ¼ Ms2 1 þ ðMp2 1ÞFn2 ðoÞ We can rearrange Eq. (5) as Ms2 1 ¼ ðMp2 1ÞFn2 ðoÞ:
ð6Þ
Table 1 Gain polynomials of some classical filters Approximation
Gain polynomial 2=n
Sync-tuned Bessel Butterworth MFMBO (n even) MFMBO (n odd) Chebyshev Ultraspherical
½1 þ ðMp
1Þo2 n 1
2=n Mp
1
on Bn ð1=oÞ on o2 b2 n 1 b2 ðo2 b2 Þn þ b2n ð1 b2 Þn þ b2n coshðn cosh1 ðoÞÞ Fna ðoÞ
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The filter order n must therefore satisfy Fn2 ðoÞ
M2 1 ¼ s2 Mp 1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ms2 1 or Fn ðoÞ ¼ Mp2 1
ð7Þ
which is one equation in four variables ðMp ; Ms ; o; nÞ for any filter polynomial F : Therefore, there are 4 1 ¼ 3-degrees-of-freedom in Eq. (7). Defining any three variables determines the fourth. This allows a virtually infinite number of parameter combinations to be used. This complicates the design process but gives enormous flexibility in adjusting filter specifications. We see in Table 1 that the filter polynomials Fn are nonlinear functions of n: Therefore, Eq. (7) is usually nonlinear and sometimes difficult if not impossible to solve in closed form. But such nonlinearities are simple to accommodate and solve using nomographs. Let us consider several different standard filters and determine their orders. From Table 1, the Butterworth filter polynomial equals on so Eq. (7) gives o2n ¼
Ms2 1 : Mp2 1
Solving for the Butterworth filter order n yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log ðMs2 1Þ=ðMp2 1Þ n¼ : log o
ð8Þ
ð9Þ
Since n is seldom an integer, we round up Eq. (9) to the next highest integer. The Chebyshev filter polynomial equals Tn ¼ coshðn cosh1 oÞ from Table 1. Using Tn as the left-hand side of the square root of Eq. (7) gives sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ms2 1 coshðn cosh1 oÞ ¼ : ð10Þ Mp2 1 Solving for the Chebyshev filter order n results in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosh1 ðMs2 1Þ=ðMp2 1Þ : ð11Þ n¼ cosh1 o These Butterworth and Chebyshev examples show that when closed-form solutions are available, their functional form is too complicated to visualize parameter interactions. Some filters use tabulated polynomials Pn ðoÞ that have been determined iteratively. Others use tabulated roots from which the polynomials may be computed. For example the catenary, parabolic, and elliptic contour filters fall in this category. Such gain functions can be expressed as 1 1 ¼ ; ð12Þ jHðoÞj2 ¼ Pn ðo=o0 ÞPnn ðo=o0 Þ 1 þ ðMp2 1ÞFn2 ðoÞ Q where Pn ðoÞPnn ðoÞ ¼ ni¼1 ðpi pni þ ðpi þ pni Þo þ o2 Þ; pi are the filter poles, and o0 is the normalized frequency for setting the passband as we shall see. The polynomials of Eq. (12) must be frequency scaled so that Pn ð1=o0 ÞPnn ð1=o0 Þ ¼ Mp2 : Solving
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Eq. (12) for the filter polynomial yields Fn2 ðoÞ ¼
Pn ðo=o0 ÞPnn ðo=o0 Þ 1 Mp2 1
ð13Þ
but it usually has no recognizable or well-known form. Nevertheless, Fn can be used for the nomograph construction. An example of this type of filter is the synchronously tunedq(sync-tuned) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi filter 2=n
which has Pn ¼ ½1 þ ðo=o0 Þ2 n=2 : All of its poles lie at o0 ¼ 1= Mp 1 so that Pn ð1Þ ¼ Mp : Therefore, equating Eqs. (13) and (7), the filter order must satisfy ½1 þ ðMp2=n 1Þo2 n 1 ¼ Ms2 1:
Rearranging Eq. (14) and solving for o gives vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2=n uM s 1 : o ¼ t 2=n Mp 1
ð14Þ
ð15Þ
There is no closed-form solution for n but it must be iteratively solved from Eq. (15).
3. General gain nomograph The general gain nomograph is constructed from Eq. (7) after it is converted into a linear equation by taking the log of both sides as [5] logðMs2 1Þ logðMp2 1Þ ¼ log Fn2 ðoÞ:
ð16Þ
By convention, a parameter g is defined as gn ðoÞ ¼ log Fn2 ðoÞ:
ð17Þ
Eq. (16) is a four-parameter equation involving ðMp ; Ms ; o; nÞ: The nomograph consists of two vertical axes labeled Mp and Ms ; and a graph of g versus o for various n: The Mp -axis and Ms -axis are the same for all filters and related to the gaxis through the basic determinant. What differs are the gn ðoÞ graphs. The axis construction of nomographs has been well described [5,6] and the nomograph has the structure shown in Fig. 1. All that is needed is to define a new filter and nomograph with Fn ðoÞ and to plot the resulting gn ðoÞ using Eq. (17). What is noteworthy about the resulting nomographs is the ease with which it is possible to relate all variables. The procedure for determining filter order begins by first expressing the required frequency response in terms of the pertinent parameters: (a) Mp ¼ the maximum attenuation (dB) in the passband. (b) Ms ¼ the maximum attenuation (dB) in the stopband. (c) Os ¼ fs =fp ¼ the normalized stopband frequency. The value of Mp and Ms are entered on the corresponding axes. A line is drawn from the point on the Mp curve through the Ms point until it strikes the g-axis. A horizontal line is then drawn across the graph corresponding to the constant g value.
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Fig. 1. (a) Basic gain nomograph showing axes and supports. (b) Determining filter order from frequency requirements.
A vertical line is drawn through the normalized stopband frequency Os : The horizontal and vertical lines intersect at some point in the graph. The minimum required order is given by the first curve lying above that point as shown in Fig. 1b. Since the nomograph effectively captures all frequency requirements in a graphical form it is easy to trade off parameters to achieve some desired characteristic. This has been treated successfully in [6] and will be expanded in Section 9 for transitional filters. It will become evident that some classical filters display transitional characteristics of their own. However, when the optimization technique is implemented for transitional filters, the relative characteristics of each filter is emphasized and then combined. Because the nomographs used here are general in form, they can be used to optimize transitional filter responses.
4. Transitional filters We have already discussed some of the classical filters and their Fn in Table 1. Now we consider transitional filters based on two classical filters having polynomials F1ðnkÞ of order ðn kÞ and F2k of order k; respectively. The transitional filter gain is
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defined as jHðoÞj2 ¼
1 1þ
e2 F12ðnkÞ ðoÞF22k ðoÞ
:
ð18Þ
The polynomial product has order n so the order of the transitional filter is maintained. The k=n ratio controls the transition between the two limiting filters where 0pk=np1: When k=n ¼ 0; the filter is an F1 type; when k=n ¼ 1; the filter is an F2 type. Setting k=nD1=2 gives the most intermediate filter. We anticipate that this transitional filter will in some manner combine the complex gain responses (i.e. magnitude jHj; phase arg H; delay d arg H=do) of the two filters. For example, the maximally flat magnitude response of the Butterworth filter combined with the equiripple magnitude response of the Chebyshev filter should produce a transitional filter with less inband gain ripple and less delay peaking. Another example is to combine the maximally flat-magnitude Butterworth filter with the maximally flat-delay response Bessel filter. We expect the transitional filter to have a less selective magnitude response and less peaking in the delay response. Several such combinations have been proposed in the past and discussed. Our purpose now is to describe the nomographs of transitional filters and their construction, pointing out the possible combinations of responses. Applying Eqs. (1) and (17) analogy to the transitional gain of Eq. (18), then the nomograph parameter equals gn ðoÞ ¼ log½F12ðnkÞ ðoÞF22k ðoÞ ¼ log F12ðnkÞ ðoÞ þ log F22k ðoÞ ¼ g1ðnkÞ ðoÞ þ g2k ðoÞ:
ð19Þ
This is a most simple but powerful result. Any nth-order transitional filter having the form of Eq. (18) is made up of two filters of order ðn kÞ and k; respectively. In terms of nomographs, its gn curve is simply the sum of its two composite g1ðnkÞ and g2k curves. These two individual nomograph curves are added together to produce the nomograph curve for the transitional filter. The transition methods used in Eq. (18) results in the simplest nomograph construction. For constant n; then k can be chosen to equal any value in the range 0pkpn: Therefore, there are ðn þ 1Þ pairs to use in Eq. (19). These pairs are: ðn; 0Þ; ðn 1; 1Þ; y; ð1; n 1Þ; ð0; nÞ: These pairs result in ðn þ 1Þ different transitional filters of fixed order n: These can be plotted on the same nomograph to provide a ready reference for transitional filters.
5. Design examples Let us now demonstrate the nomograph construction with several examples. We begin by considering the transitional Butterworth–Chebyshev filter, with gain jHðoÞj2 ¼
1 1þ
e2 o2ðnkÞ Tk2 ðoÞ
:
ð20Þ
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The g parameter therefore equals gn ðoÞ ¼ logðo2ðnkÞ Þ þ logðcoshðk cosh1 ðoÞÞ2 Þ:
ð21Þ
Let us consider the case where order n ¼ 5: The six different order combinations ðn k; kÞ are ð5; 0Þ; ð4; 1Þ; ð3; 2Þ; ð2; 3Þ; ð1; 4Þ; and ð0; 5Þ: The individual Butterworth and Chebyshev nomographs are shown in Fig. 2. If we choose n ¼ 0–5 curves and pair them as above, we can employ Eq. (21) and plot the resulting nomograph as shown in Fig. 3. The six different TBC filters have the same asymptotic slope (d log jHj=d log o ¼ 5) for o-10 but different curvatures from the nomograph origin at o ¼ 1: These curvatures are more than Butterworth but less than Chebyshev except at the extremes. This example shows the combination of a high-selectivity high-delay distortion Chebyshev filter with a moderate-selectivity moderate-delay Butterworth filter. The result is a compromise filter that has higher selectivity than the Butterworth filter but lower delay distortion than the Chebyshev filter. 2=n Another example is the synchronously tuned filter which has Pn ¼ ð1 þ ½Mp 2 n=2 1o Þ : Normally we would use a Bessel filter but we cannot write a closed-form expression for its gain. The sync-tuned and Bessel filters have fairly similar magnitude and delay characteristics, so we can substitute the sync-tuned to demonstrate the process. Using the Butterworth polynomial, the g parameter therefore equals gn ðoÞ ¼ logðo2ðnkÞ Þ þ logð½1 þ ½Mp2=k 1o2 k Þ:
ð22Þ
This TBS nomograph is shown in Fig. 4. Here we have combined two nomographs, one with straight lines (Butterworth) and one concave downward (sync-tuned). This combines a moderate-selectivity moderate-delay-distortion Butterworth filter with a low-selectivity low-delay-distortion sync-tuned filter. The produces a lower selectivity filter with lower delay distortion. Still another example is the Chebyshev–sync-tuned transitional filter. Its characteristics will be very close to a Chebyshev–Bessel transitional filter. The g parameter for this combination is gn ðoÞ ¼ logðcoshððn kÞ cosh1 ðoÞ2 Þ þ logð½1 þ ½Mp2=k 1o2 Þk Þ:
ð23Þ
This transitional Chebyshev–sync-tuned (TCS) nomograph is shown in Fig. 5. The resulting nomograph shows the combination of one curve that is concave upward (Chebyshev) with another that is concave downward (sync-tuned). Therefore, we are combining a high-selectivity high-delay-distortion Chebyshev filter with a lowselectivity low-delay-distortion sync-tuned filter. Evidently, we can obtain an intermediate filter that has a nomograph close to that of a Butterworth filter (cf. k ¼ 3 or 4 with Fig. 2b). We can also present some recent examples of nontransitional filters that can be converted into transitional filters. A Butterworth-inverse–Chebyshev transitional filter [23] would have gn ðoÞ ¼ logðo2ðnkÞ Þ þ logðcoshðk cosh1 ðos =oÞÞ2 Þ;
ð24Þ
where os is the normalized stopband frequency and the inverse Chebyshev formulation is from [24]. A MFMBO–Chebyshev transitional filter
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Fig. 2. Gain nomographs for (a) sync-tuned, (b) Butterworth, and (c) Chebyshev.
would have gn ðoÞ ¼ log
2 nk o b2 þlogðcoshðk cosh1 ðoÞÞ2 Þ 1 b2
ð25Þ
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Fig. 3. Nomograph of fifth-order transitional Butterworth–Chebyshev filter.
for n even and gn ðoÞ ¼ log
b2ðnkÞ þ ðo2 b2 Þnk b2ðnkÞ þ ð1 b2 Þnk
! þ logðcoshðk cosh1 ðoÞÞ2 Þ
ð26Þ
for n odd. The parameter b is selected for providing maximally flat magnitude slope condition at frequency 0pbp1 and is a shaping control. We note in passing that any number m of filters can be transitionally combined as jHðoÞj2 ¼
1þe
1 Q 2
m
Fm2 i
;
X
mi ¼ n
in which case the nomograph parameter equals X X gðoÞ ¼ logðFm2 i Þ mi ¼ n:
ð27Þ
ð28Þ
m
Here the transitional nomograph is always the average of m individual nomographs.
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Fig. 4. Nomograph of fifth-order transitional Butterworth–sync-tuned filter.
6. Transitional filters having alternative forms The second standard form for transitional filter gain involves interpolating the poles and zeros of two filters, denoted as s1 and s2 ; respectively. One interpolation is Form A :
jsi j ¼ js1i jm js2i j1m ;
yi ¼ my1i þ ð1 mÞy2i ;
ð29Þ
where y1i ¼ argðs1i Þ and y2i ¼ argðs2i Þ: The root magnitudes are geometrically interpolated and the root angles are arithmetically interpolated. The transition parameter 0pmp1: Taking the logarithm of the magnitude shows log jsi j ¼ m log js1i j þ ð1 mÞ log js2i j
ð30Þ
and that the magnitude interpolation is geometric. Another magnitude interpolation that can be employed is Form B :
jsi j ¼ mjs1i j þ ð1 mÞjs2i j;
yi ¼ my1i þ ð1 mÞy2i :
ð31Þ
In this case, the root magnitudes are arithmetically interpolated like the phases. Other interpolation schemes like geometrically interpolated angle can be considered (although they have not been proposed in the past) but all the results are processed as follows:
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Fig. 5. Nomograph of fifth-order transitional Chebyshev–sync-tuned filter.
The transitional filter transfer function HðsÞ is then k k ¼ Qn : HðsÞ ¼ Qn jyi i¼1 ðs þ si Þ i¼1 ðs þ jsi je Þ In ac steady state, s ¼ jo and the squared gain equals k jHðoÞj2 ¼ Qn : 2 2 ðjs j þ 2ojs i i j sinðyi Þ þ o Þ i¼1
ð32Þ
ð33Þ
Therefore, this gain takes the polynomial form of Eq. (12). The gain polynomial must then be determined using Eq. (13) which yields n Y Fn2 ðoÞ ¼ ðjsi j2 þ 2ojsi j sinðyi Þ þ o2 Þ 1: ð34Þ i¼1
Using the geometric interpolation form of Eq. (29), this gain polynomial becomes n Y Fn2 ðoÞ ¼ ðjs1i j2m js2i j2ð1mÞ þ 2ojs1i jm js2i j1m i¼1
sinðmy1i þ ð1 mÞy2i Þ þ o2 Þ 1:
ð35Þ
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Using the other arithmetic interpolation form of Eq. (31), this gain polynomial becomes n Y ðm2 js1i j2 þ ð1 mÞ2 js2i j2 þ 2mð1 mÞjs1i j js2i j Fn2 ðoÞ ¼ i¼1
þ 2oðmjs1i j þ ð1 mÞjs2i jÞ sinðmy1i þ ð1 mÞy2i Þ þ o2 Þ 1:
ð36Þ
Unfortunately, these polynomial forms are usually not recognizable functions as in the case of Eq. (1). Therefore, the gain nomographs are plotted directly using Eqs. (35) and (36), or some other resulting interpolation formula. These interpolations do not result in nomographs that are sums of constituent nomographs. Another comment should be made. The roots used in the interpolations of Eq. (29) or Eq. (31) are usually normalized so the two respective filters have unit passband bandwidth. When the root magnitudes are interpolated, the resulting transitional filter generally does not have unit bandwidth but bandwidth o0 : Therefore, the root magnitudes must be reduced or increased by the frequency scaling factor o0 as m 1m s1 i s2 i s1i s2 ð37Þ jsi j ¼ or jsi j ¼ m þ ð1 mÞ i o0 o0 o0 o0 before constructing the gain nomograph using Eq. (35) or Eq. (36). The root angles are unchanged. To demonstrate these interpolation methods, reconsider the Butterworth–Chebyshev fifth-order transitional filter discussed in Eq. (21). The Butterworth poles are s1 ¼ 1:0000;
js1 j ¼ 1:0000;
y1 ¼ 01;
s2 ; s3 ¼ 0:80907j0:5878;
js2 j; js3 j ¼ 1:0000;
y2 ; y3 ¼ 7361;
s4 ; s5 ¼ 0:30907j0:9511;
js4 j; js5 j ¼ 1:0000;
y4 ; y5 ¼ 7721:
ð38Þ
These angles are with respect to the negative real axis. The Butterworth poles lie on a unit circle with angles that are multiples of 1801=5 ¼ 361: The Chebyshev poles are s1 ¼ 0:1775;
js1 j ¼ 0:1775;
y1 ¼ 01;
s2 ; s3 ¼ 0:14357j0:5970;
js2 j; js3 j ¼ 0:6140;
y2 ; y3 ¼ 776:51;
s4 ; s5 ¼ 0:05497j0:9659;
js4 j; js5 j ¼ 0:9762;
y4 ; y5 ¼ 786:71:
ð39Þ
The poles lie on an ellipse having a semi-minor axis length of 0.1775 and a semimajor axis length of one. Both sets of poles given by Eqs. (38) and (39) have been adjusted to produce 3 dB passband ripple and unit ripple bandwidth. Since o0 ¼ 1; no frequency scaling of roots is needed as discussed in Eq. (37). Consider the geometric interpolation of Eq. (29). The TBC filter poles are js1 j ¼ 1m 0:17751m ;
y1 ¼ 01;
m
1m
;
y2 ; y3 ¼ 7½m361 þ ð1 mÞ76:51;
m
1m
;
y4 ; y5 ¼ 7½m721 þ ð1 mÞ86:71:
js2 j; js3 j ¼ 1 0:6140 js4 j; js5 j ¼ 1 0:9762
ð40Þ
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The pole locations move from the Chebyshev ellipse (m ¼ 0) to the Butterworth circle (m ¼ 1) as m is increased from zero to unity as shown in Fig. 6a. The five root loci are almost horizontal so the imaginary part of the poles are almost constant. The resulting gain nomograph is shown in Fig. 6b. The intermediate filters occur for mD1=2: Comparing the TBC filter given by Eq. (21) whose nomograph is shown in Fig. 3, we see little difference. If we label the curves of Fig. 3 as k=n ¼ ð0; :1; :2; y; 1Þ; then mDk=n in Fig. 6b. Now consider the arithmetic interpolation of Eq. (31). The TBC filter now has js1 j ¼ m þ ð1 mÞ0:1775;
y1 ¼ 01; y2 ; y3 ¼ 7½m361 þ ð1 mÞ76:51; y4 ; y5 ¼ 7½m721 þ ð1 mÞ86:71:
js2 j; js3 j ¼ m þ ð1 mÞ0:6140; js4 j; js5 j ¼ m þ ð1 mÞ0:9762;
ð41Þ
The pole angles of Eq. (41) are the same as those used in Eq. (40). The root locus is shown in Fig. 7a. The pole vector lengths shorten by equal amounts while the angles also decrease by equal amounts. Comparing this root locus with that of the other interpolation shown in Fig. 6a, we see that path is more curved. This slight difference is minor in the magnitude responses, more noticeable in the phase responses, and particularly noticeable in the delay responses. There is negligible difference in the impulse and step time domain responses. The nomograph is shown in Fig. 7b. Again mD1=2 gives an intermediate response. There is little difference between Figs. 6 and 7 so mDk=n: To show more dramatic differences, reconsider the Chebyshev–sync-tuned fifthorder transitional filter discussed in Eq. (22). The sync-tuned poles are s1 ; s2 ; s3 ; s4 ; s5 ¼ 2:593;
js1 j; js2 j; js3 j; js4 j; js5 j ¼ 2:593;
y1 ; y2 ; y3 ; y4 ; y5 ¼ 01:
ð42Þ
All poles are located at 2:593: The 3 dB filter bandwidth is unity. The Chebyshev poles were given in Eq. (39) and also have unit 3 dB bandwidth. Using the geometric interpolation of Eq. (29), the TCS filter poles are js1 j ¼ 2:593m 0:17751m ;
y1 ¼ 01;
m
1m
;
y2 ; y3 ¼ 7½m01 þ ð1 mÞ76:51;
m
1m
;
y4 ; y5 ¼ 7½m01 þ ð1 mÞ86:71:
js2 j; js3 j ¼ 2:593 0:6140 js4 j; js5 j ¼ 2:593 0:9762
ð43Þ
The root locus is shown in Fig. 8a and the nomograph in Fig. 8b. The arithmetic interpolation of Eq. (31) gives a TCS filter with poles js1 j ¼ m2:593 þ ð1 mÞ0:1775;
y1 ¼ 01;
js2 j; js3 j ¼ m2:593 þ ð1 mÞ0:6140;
y2 ; y3 ¼ 7½m01 þ ð1 mÞ76:51;
js4 j; js5 j ¼ m2:593 þ ð1 mÞ0:9762;
y4 ; y5 ¼ 7½m01 þ ð1 mÞ86:71:
ð44Þ
The root locus is shown in Fig. 9a and the nomograph in Fig. 9b. The geometric interpolation gives a flatter root loci than the arithmetic interpolation. But both loci have considerably more arc than the earlier TBC filter. Both nomographs have more curvature which reflects the root loci arcs. The TCS filter plots of Fig. 5 using
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Fig. 6. Fifth order transitional Butterworth–Chebyshev filter with geometric interpolated poles. (a) Root locus and (b) nomograph.
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Fig. 7. Fifth order transitional Butterworth–Chebyshev filter with arithmetic interpolated poles. (a) Root locus and (b) nomograph.
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Fig. 8. Fifth order transitional Chebyshev–sync-tuned filter with geometric interpolated poles. (a) Root locus and (b) nomograph.
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Fig. 9. Fifth order transitional Chebyshev–sync-tuned filter with arithmetic interpolated poles. (a) Root locus and (b) nomograph.
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polynomial interpolation rather than root interpolation are fairly similar so mDk=n: However the root locus spans a much larger area and is largely circular unlike the TBC filters of Figs. 6 and 7. 8. Transitional filters using transmission zeros We now consider transitional filter nomographs that arise using k pairs of transmission zeros. The gains of such filters often take the form 1 jHðoÞj2 ¼ ; k=2on; ð45Þ Qk=2 2 2 2 1 þ e Fn ðoÞ=c i¼1 ðo o2i Þ2 Q 2 where Fn is a filter polynomial of order n and ðo o2i Þ is the transmission zero polynomial of order k: For the filter to remain lowpass and have jHðNÞj ¼ 0; k must be chosen so ko2n: The nomographs of such equations are simple to construct numerically but are extremely difficult to evaluate analytically. Let us demonstrate this using a Butterworth filter having Fn2 ¼ o2n and one pair of transmission zeros 7o0 : Then the gain equals 1 : ð46Þ jHðoÞj2 ¼ 1 þ ð1 o20 Þ2 o2n =ðo2 o20 Þ2 The transitional filter order is n and the asymptotic rolloff rate is 20ðn 2Þ dB=dec: The dc gain jHð0Þj is unity and the 3 dB corner frequency has been normalized to unity since jHð1Þj2 ¼ 1=2 by the use of the ð1 o20 Þ2 factor. If o0 ¼ N; the filter is Butterworth. To obtain greater band-edge selectivity, the transmission zero o0 is chosen as 1oo0 oN to obtain the desired stopband rejection. The case when o0 ¼ 2 is shown in Fig. 10 for n ¼ 2–5. We see that the gain rapidly approaches zero beyond the 3 dB band-edge frequency of unity. The gain equals zero at o0 ¼ 2 and then rebounds to some value Ms beyond o0 : For plotting nomographs, only the gain portion from o ¼ 1 to os where jHðos Þj ¼ 1=Ms is used. Then Ms and os must be determined. This is done as follows: The minimum stopband rejection Ms occurs at the stopband peaking frequency opeak where djHj2 =do2 ¼ 0: Solving Eq. (46) leads to the simple result that opeak ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o0 n=ðn 2Þ: The minimum stopband rejection Ms is obtained by evaluating jHðopeak Þj ¼ 1=Ms to obtain n n n 2 2 n1 2 Ms Do0 ðo0 1Þ : ð47Þ n2 2 Then using Eq. (46), the stopband frequency os satisfies n n n 2 2 on ðo2 1Þ 2 Ms D s 2 0 2 Don1 ðo 1Þ 0 0 n2 2 ðos o0 Þ
ð48Þ
which is highly nonlinear and must be solved numerically for os : To determine the nomograph, we must find the op and Ms values for o > o0 : We then extend Ms backwards until it intersects the gain characteristic for 1oooo0 : We then plot the gðoÞ in this range. Using this technique, the Butterworth-single
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Fig. 10. Magnitude response of Butterworth filter for n ¼ 2; 3; 4; 5 and zero at o0 ¼ 2:
transmission zero nomograph is shown in Fig. 11 for o0 ¼ 2: The unused portion of the filter gain characteristic is shown by solid curves. Another useful combination of single transmission zero nomographs are shown in Fig. 12. This shows the cases for o0 ¼ 2; 3; y; 9: We see the Butterworth nomograph curves which are the straight line segments that connect the ð0; 1Þ origin to gn ¼ 4; 6; y; 20 at o ¼ 10 for n ¼ 2; 3; y; 10; respectively. The single transmission zero produces more stopband attenuation. This produces the tangentially upward curves shown in Fig. 12. The nomograph parameters gn ðoÞ equals
2 o o20 2n gn ðoÞ ¼ logðo Þ log ; n>2 ð49Þ 1 o20 by viewing Eq. (46) as having the form of Eq. (18) and the parameters of Eq. (19). Notice the negative sign of the transmission zero term in this formulation. Asymptotically, the Butterworth gain slope is 20n dB=dec while the transmission zero gain slope is 40 dB=dec: This emphasizes that transmission zeros improve (i.e. increase) the band-edge selectivity but reduce the asymptotic attenuation by 40 dB=dec: Multiple zeros can be used but they must generally be aligned by some analytical technique as with inverse Chebyshev and elliptic filters, or by some numerical technique like the Remez algorithm or alternatives [25]. Regardless of what approach is used, the resulting transmission zero function can be used to form the nomograph as described above. The nomograph parameter equals 0 1 Y 2 gn ðoÞ ¼ logðFnþk Þ log@c ðo2 o20 ÞA; kon=2 ð50Þ k=2
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Fig. 11. Nomograph of Butterworth filter with transmission zero at normalized frequency 2. Nomograph should use right-hand portion beyond inflection point.
for a 40n dB=dec rolloff filter. General gain functions with arbitrary pole and zero ripples require a different approach. Many numerical problems arise when constructing the nomograph of general gain functions. Consider the general attenuation function shown in Fig. 13(a). The attenuation ripples in both the passband (below passband frequency op ) and stopband (above stopband frequency os ). Both bands exhibit non-equiripple response. The maximum passband attenuation is Mp and the minimum stopband attenuation is Ms : These levels and frequencies are shown and labeled in Fig. 13(a). When constructing the nomograph, only the transition gain characteristic between op and os is used; the rest of the characteristic is ignored. The four parameters used in the nomograph are ðMp ; op ; Ms ; os Þ and the nomograph construction of general filters begins by solving for these parameters from four equations using numerical methods. Then Mp and Ms are used to determine g and Os ¼ os =op : The nomograph curves are plotted for fixed filter order n as a function of ðg; Os Þ for 0pgp12 and 1pOs p10:
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Fig. 12. Single transmission zero showing greater stopband attenuation initially at the expense of decreased asymptotic attenuation.
Fig. 13. (a) Definition of filter attenuation jHj2 and (b) mapping to nomograph parameter g:
We now describe the steps which enable us to determine ðMp ; op ; Ms ; os Þ: The steps involve the general filter gain which has the form Q ðjo þ pi Þðjo þ pni Þ jHðjoÞj2 ¼ 1 þ Q n ; ð51Þ n m ðjo þ zi Þðjo þ zi Þ where the denominator represent zeros of transmission when Reðzi Þ ¼ 0: The first step is to find the passband peaking frequencies o1 from djHj2 ¼ 0; 0po1 pop : ð52Þ do2 o¼o1
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In Fig. 13(a), there are five such frequencies (i.e., peaks and valleys). These values are determined by solving Eq. (52) using numerical methods. The second step is to select the maximum-peaking frequency o1 that gives the maximum passband attenuation Mp and to find the corresponding passband frequency op : This is accomplished by solving max jHðjo1 Þj2 ¼ Mp2 ¼ jHðjop Þj2 :
ð53Þ
In Fig. 13(a), the five frequencies are found from Eq. (52). These frequencies are substituted into the gain in Eq. (51). The maximum attenuation value is defined to be Mp and the frequency which produces it is o1 : After Mp is thus determined, numerical methods are used to find the frequency op that satisfies Eq. (53). This concludes the steps necessary for evaluation of the passband. The third step is to find the stopband peaking frequencies o2 where djHj2 ¼ 0; do2 o¼o2
os po2 pN:
ð54Þ
In Fig. 13(a), there are seven such frequencies (i.e., peaks and valleys) which are found by solving Eq. (54) by numerical techniques. The fourth and final step is to select the minimum-peaking frequency o2 that gives the minimum stopband attenuation Ms and to find the corresponding stopband frequency os : This is accomplished by solving min jHðjo2 Þj2 ¼ Ms2 ¼ jHðjos Þj2 :
ð55Þ
In Fig. 13(a), seven frequencies are found from Eq. (54). These frequencies are substituted into the gain in Eq. (51). The minimum attenuation value is defined to be Ms and the frequency which produces it is o2 : After Ms is thus determined, numerical methods are used to find the frequency op that satisfies Eq. (55). This concludes the steps necessary for evaluation of the stopband and results in a nomograph such as shown in Fig. 13(b). The problem with this technique is that the derivatives in Eqs. (52) and (54) are highly nonlinear equations and cannot generally be solved in closed form. The same can be said of frequencies involved in Eqs. (53) and (55). Therefore, numerical methods must be used to determine the parameters used to form the nomograph, but the end result is a graphical rendition of the desired filter characteristics. This is unnecessary for classical filter nomographs. The all-pole filters have no stopband ripple. Therefore, the derivative of step 3 using Eq. (54) is unnecessary and Eq. (55) has a closed-form nonlinear solution. In the passband, the classical filter polynomials have been chosen to be bounded by known ripple (often jFn jp1) over the range 0pop1 and monotonically increases beyond that range. Thus, the first step of finding the derivative in Eq. (52) is unnecessary and Eq. (53) is known by polynomial substitution.
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9. Optimizing transitional filters using nomographs In [6] the authors proposed a technique for optimizing classical filter magnitude response based on filter nomographs as those presented here. The optimization was the achievement of any of the following without increasing filter order: 1. Maximizing band-edge selectivity. 2. Minimizing passband ripple. 3. Maximizing stopband rejection. The method consisted of adjusting the filter requirements to obtain a new set of poles/zeros for the transfer function. The resulting filter used all degrees of freedom (except order) to achieve the desired optimization. The filter nomographs considered here effectively capture all requirements (degrees of freedom) for the design. Since the order must be an integer, any ‘‘excess’’ order is available for optimization. For optimizing the frequency response of a filter as stated above, the values of Mp ; Ms and Os can all be determined readily from the nomograph. In effect, it is possible to move the point in Fig. 1b along a constant g or Os line to the curve corresponding to the required order. This effectively performs the iteration in Eq. (16) for all values depending on Fn : From this optimization technique we can note that several classical filters can be made to have ‘‘transitional’’ characteristics. For example, the elliptic filter can be ‘‘optimized’’ for less delay variation by minimizing passband ripple for the given order. Another option is to reduce passband delay peaking by minimizing selectivity (maximizing transition bandwidth). For Chebyshev filters, passband ripple can be minimized to reduce delay distortion. The degrees of freedom available for optimization can be noted from the selectivity equations as described in [26]. For the case where the transitional filter consists of a combination of two responses, the result is a new Fn that is captured in the nomograph. Two approaches become apparent: (a) Optimize the individual responses prior to combining them in the transitional form. This requires that the engineer know the order of the filter prior to implementation, which can be obtained from the transitional nomograph. (b) Optimize the response directly from the transitional nomograph. This is the most straight-forward way: The requirements are entered in the nomograph and the filter order determined. The optimization described in [6] can then be performed on the transitional nomograph: adjusting Mp ; Ms or Os : In the case of transitional filters implementing pole/zero interpolation, the method is more involved. Here, each response must be treated independently and the corresponding poles and zeros calculated. The interpolation is then implemented to achieve the desired combination. The transitional nomograph for interpolated poles/ zeros can be used to determine the order in most of the cases.
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As an example, let us consider the following filter requirements: Mp ¼ 1:25 dB; Ms ¼ 40 dB; and Os ¼ 3: In either case we could implement a fifth-order transitional Butterworth–sync-tuned or Chebyshev–sync-tuned. For the Butterworth filter, the only degree of freedom is order, so let us choose the Chebyshev–sync-tuned filter for optimization. Entering the passband and stopband requirements into Fig. 5 we get k ¼ 4 for the Chebyshev response. We note that we have some margin because there is excess rejection. We can maximize selectivity by reducing the stopband corner ratio to Os ¼ 2:5: We can also reduce the passband ripple by selecting where the k ¼ 4 curve intersects Os ¼ 3 and reading gE5:3: Drawing a line from this point through Ms ¼ 40 dB gives us a new passband ripple requirement of Mp ¼ 0:2 dB: The advantage of doing this is the reduction of delay ripple variation in the Chebyshev passband, further emphasizing delay response of the transitional filter.
10. Conclusions We have presented the construction of transitional filter nomographs. For the case where two (or more) filter functions are combined, the resulting nomograph is the sum of the individual filter nomographs. This simple yet powerful result means that no additional theoretical work needs to be completed when two potential filters are contemplated to be transitionally combined! Since an nth-order transitional filter can be realized in ðn þ 1Þ different forms using the ðn; 0Þ; ðn 1; 1Þ; y; ð0; nÞ pairs, there are a large varieties of possibilities to be considered. The nomograph can therefore be sketched using the ðn þ 1Þ pairs of curves without resorting to computers, etc. Optimization of these combined responses was also described. For alternative forms using pole and zero interpolation, the nomographs are constructed directly from the interpolation formulas. In certain cases, the resulting transitional filter magnitude response are similar to the combined function case. In general, however, the pole interpolation significantly distorts phase/delay properties, so they merit further investigation. The purpose here was to identify filter transition forms and generate nomographs based on those forms for ease of filter selection and design. With these nomographs and the proposed optimization technique, the engineer can make ready evaluation of the unique capabilities of transitional filters.
References [1] M. Kawakami, Nomographs for Butterworth and Chebyshev filters, IEEE Trans. Circuit Theory CT10 (1963) 288–290. [2] E. Christian, E. Eisenmann, Filter Design Tables and Graphs, Wiley, New York, 1966. [3] A.I. Zverev, Handbook of Filter Synthesis, Wiley, New York, 1967. [4] C.S. Lindquist, Active Network Design with Signal Filtering Applications, Steward & Sons, CA, 1977 (Chapter 4). [5] J.N. Hallberg, C.S. Lindquist, Nomographs and filters, J. Franklin Inst. 302 (1976) 111–145.
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