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Nomographs and filters
Journal of The Franklin DEVOTED
Volume
302,
Number
TO SCIENCE
AND
Institute THE MECHANIC
ARTS
August
2
1976
Nomographs and Filters by JOE N. HALLBERG
Interstate Electronics Corporation, 707 E. Vermont, Anaheim, California
and
CLAUDE
s. LINDQUIST
Department of Electrical Engineering, California State University, Long Beach,
California
AEISTRACKThis paper presents the nomographs for additional classical filters, including ultraspherical, Legendre, modified associated Legendre, Papoulis, Halpern, Bessel, Gaussian, and synchronously-tuned. It also identifies inaccuracies in the earlier nomographs. The basic theory of nomographs and their utilization in developing filter nomographs is presented.
I. Introduction Nomographs have been used extensively for a half century in the sciences for graphically representing mathematical relationships. In 1963, Kawakami applied nomography to the area of classical lowpass filter response (1).He introduced three nomographs which described Butterworth, Chebyshev, and elliptic filters. These nomographs depicted the interrelationship between the maximum passband attenuation Mp, the minimum stopband rejection MS, the normalized stopband frequency R, and the filter order n as shown in Fig. 1. Given any three of these variables, the fourth could be easily determined by inspection. As an example, if MP = 1 dB, MS = 40 dB, and Sz = 1.5, then the order n of the filter will be obtained as the figure demonstrates. This capability is invaluable in filter design. Later in 1967, similar nomographs were published by Zverev (2) and have become widely used by industry in passive and active filter design [for an alternative form, see (3)]. Unfortunately, with the myriad of other filter types commercially available and in use (e.g. Papoulis, Bessel, etc.), no additional nomographs have emerged. Not only has this hindered rapid design, but more importantly, it has hampered easy comparison of the various filter types. This, coupled with the fact that the nomographs of (1)and
111
J. N. Hallberg and Claude S. Lindquist
FIG. 1. (a) Typical
magnitude
response
of lowpass filter and (b) nomograph nitude response.
of mag-
(2) contain inaccuracies, has motivated the study of nomographs and filters presented in this paper (4). This paper addresses and discusses four major topics: the general theory of nomographs, application of nomography to filters, the classical filter responses which are widely known, and the nomographs which describe these responses. It will be shown that the theoretical basis of filter nomographs is rather simple. Yet most engineers find it surprising that so much filter information can be obtained from them. These nomographs provide truly marveilous design aids as we shall see. We will also present a design example, and identify inaccuracies in the earlier work. 112
Journal of The
Franklin
Institute
Nomographs and Filters 5-
10
5
/ /' /'
" u
I' '6
3p
"3
/' /' 1' /'
2L
4
2
2
4--
FIG. 2.
Nomograph
0/8
0 i
for u + 2)= w. Given u = 2 and u = 4, then w = 6.
We should also note the generality of this work. The nomographs to be constructed will describe the magnitude response of lowpass filters. By use of the Ibwpass-to-highpass, lowpass-to-bandpass, and lowpass-to-bandstop transformations (5), they can equally well be applied to these other classes of filters. II. General
Theory of Nomographs
Nomographs derive their name from the Greek words nomos (law) and graphein (to write). They are literally graphical representations of mathematical laws or relationships. Nomography, the study and use of nomographs, was introduced in 1891 by Lt. Col. M. d’ocagne, the French mathematician, in his “Traite De Nomographie” (67). Nomography quickly spread to England, Germany, and other parts of the world. The Germans refer to them as nomograms and the English call them nomographs. Occasionally, they are called alignment charts. In this paper, they will be referred to as nomographs. A nomograph is simply a graphical representation of equations in three or more variables. We will first describe a nomograph with three variables. Each variable is plotted on a line. Given any two, the third can be obtained. Drawing a straight line between the given two, the intersection on the third line gives the value of the third variable. An example of a nomograph is shown in Fig. 2. We shall now discuss the theoretical basis for nomographs. d’Ocagne based his development of nomographs upon projective geometry and the principle of duality, using parallel line coordinates (6). We will use the approaches of Allcock (8) and Epstein (9) that use determinants and Cartesian coordinates. The most fundamental principle of all nomographs is that any three points, say (xi, yr), (x2, y2), and (x3, y3), lie on the same straight line when the value of the determinant x1
Vol. 302, No. 2. AugustI!?76
Yl
1
x2
y2
1
x3
y3
1
(1)
113
J. N. Hallberg
and Claude
S. Lindquisl
xI FIG. 3.
x
5
x2
Three functions u, v, and w.
is zero (10). This can easily be shown to be true (9) by letting u, u), and w, denote the three functions on which points (x1, yi), (x2, yz), and (x3, y3) lie as shown in Fig. 3. Then let A, B, and C be the intersections of an arbitrary straight line with the u, 21,and w curves, respectively. Denote point (x2, yi) by D and (x3, y2) by E. Thus, AD = x2-x1 and DB = y2- yl. Similarly, EC= y3 - y2 and BE = x3 - x2_ Equating the hypotenuse slopes of triangles ABD and BCE, we see y3-y2 y2-=
Clearing
the fractions
(2)
x3-x2’
x2-x1
and rearranging,
we obtain (3)
~ly2+~2y3+~3yl-~ly3-~2yl-~3y2=0~
This may be written
in determinant
notation
as
Xl
y1
1
x2
y2
1 =o.
I x3
y3
1
(4)
I
Now, if xi and yi are functions of u only, x2 and y2 of v only, and xg and y3 of w, we can rewrite the determinant as &5(u)
f,(u)
I
g2(u)
f2(u)
1
g3(w)
f3(w)
1
=o.
(5)
This determinant is often called the basic determinant. If any equation can be written so it is possible to form the basic determinant, then it is possible to construct a nomgraph consisting of three curves in the (x, y)-plane which are graduated and represent the variables u, v, and w (8). 114
Journalof The
Franklin Institute
Nomographs and Filters
It is important to re-emphasize that the basic determinant of any nomograph must satisfy the following conditions: (a) The value must be zero. (b) Each row must contain one variable only. (c) The last element of each row must be positive and equal to unity. As we shall now discuss, the nomograph is formed from the basic determinant by plotting the points [gl(u)=xl, f~(u>=~11, [gdu) =x2, Mu)= ~21, and [&(W)= x3, f3(w) = Y31,for various values of U, u, and w. This produces the lines of the nomograph as shown in Fig. 2. The first step in designing a nomograph is the formation of the basic determinant. The general method of deriving the basic determinant from any formula containing three variables consists, essentially, in forming three simultaneous equations of the form Ax +By + C = 0. From these equations, the determinant
i
AI
6
G
A2
B2
C,
A3
B3
G
=0
(6)
I
may be derived directly by writing the coefficients of the equations in the order shown. Then this determinant is transformed into the basic nomographic form by taking advantage of any of the well-known determinant manipulations. We now illustrate the formation of the basic determinant with an example (11). We design the nomograph shown in Fig. 2 for the equation u+v=w
(7)
since this equation will be utilized later in filter design. First, we rewrite the equation as u+v-w=o.
(8)
Then we introduce the parametric equations x = u and y = 2). With these two equations and the one above, we have the required three where x-u=0 y-v=0
(9)
x+y-w=o. Substituting these equations into the determinant
I 01
01
-w -u -v
I
=o.
form of (6), we obtain
(IO)
It now remains to express this determinant as a basic determinant in the standard form of (5). This is accomplished by using basic determinant manipulations as will now be described. In the above determinant, column 1 is replaced by the sum of columns 1 and Vol. 302, No. 2, August 1976
115
J. N. Hallberg
and
Claude
S. Lindquist
2, and column
3 is multiplied
by -1,
i 21 The bottom
row is divided
yielding
w u2, I =o.
01
by 2; resulting 10
By interchanging
columns,
(11)
in u
1 I 1
1
z1
1
t
WI2
=o.
the final form is achieved
I
0
ul
1 1
v
2
w/2
(12) as
1 =o
(13)
1I
which is the basic determinant. Note that all the steps shown above were permissible because the value of the determinant was zero. The nomograph of Fig. 2 is constructed from this basic determinant. Comparing this basic determinant to (4), the equations are:
The nomograph w, and v at
xi = 0
y1=u
x2=1unit
y2=v
x3 = 4 unit
y3 =
is constructed
by first drawing
(14) w/2. the three
“scale supports”
for u,
xr=o x3=+
(I%
x2= 1. Then
the “functional
scales”
are drawn for u, w, and v. The functional scale are marked with the “value of the variable” u, w, or v and on which the graduations are laid off in proportion to the corresponding values of the “function of the variable” f(u), f(w), or f(v). The distances are laid off from an initial point of the scale, which are not necessarily the zero points. Although it is not necessary in this simple example, tables are often used to describe the functional scales. Defining u and v to vary from 0 to 5 (cm) and w to vary from 0 to 10 (cm), then f(u) and f(v) are listed in Table I(a) and f(w) in Table I(b). The functional scales are then drawn in Fig. 4. This completes the nomograph for u + 2) = w. When the function values, f(u), f( v ) , or f(w), do not lie in the proper range is used. The scale modulus m is a scale for plotting, a “scale modulus” multiplier which enables us to place the value of a function within a prescribed range (12), (13). Mathematically, the modulus m equals
(11)is one on which graduations
(16)
116
Journal of The Franklin Institute
Nomographs and Filters TABLE Functional
I
scales for Fig. 4
where x is the length, fHand fL are the values of the functions at the upper and lower limits of U, and (fH- fL) is the range of the function (13). This is one of the most useful equations in nomography. For example, consider the equation u + u = w once again. However, this time assume that u varies from 2 to 10 and that II varies from 5 to 15. If the lengths of the u and v are scales of 6 cm, then the scale moduli are
mu===;
6
3
6 m, =15_5=?.
(17)
3
Now let us see the effect these scale moduli have on the nomograph
FIG.~. Nomograph
for u+u=
w.
of Fig.
J. N. Hallberg and Claude S. Lindquist 4. Scaling u and v by scale moduli m,, and m, in (9) gives x - m,,u = 0 y-m,v=O
(18)
x/m, + y/m” - w = 0.
We can reform determinant
(10) as
-m,u 0 -r&v =o. 0 1 l/m, l/m, -w is easily reduced to the basic determinant
1
This determinant
0 1 mJ(m, + mv)
m,u r&u m,nkw/(mU + mu)
(19) form as
1 1 =o
(20)
1
For m, =a and m, = 2, then (20) equals (21) To draw the nomograph, we adjust the functions of u, v, and w so that the zero of measurement (i.e. y = 0) coincides with the first values of the functions (11). This usually requires simple translation of u, v, and w. Translating u, v, and w in (21), then 10 3(u-2)/4 11 1 3(v-5)/5 1 =o. (22) d (w-7)/3 1 We can now form a table of the functional scales of the nomograph as shown in Table II. From Table II and the x coordinates in the basic determinant (22), the nomograph is constructed in Fig. 5. The basic determinant (20) is the general determinant of the equation u + v = w. It reduces to (13) by letting m, = 1 and m, = 1. Now let us apply these results to filter analysis. III. Applying Nomography
to Filters
The transfer function H(s) of an nth order lowpass filter equals 1 N,(s) H,(s)=-=M”(S) D,(s)’ m< n where A4 denotes its reciprocal, and N and D are polynomials filters have squared magnitude responses of lH(jwV=
(23)
in s. Classical (24)
,Mfjw),2 = 1 + E21si,(w)
where S,(w) is related to a classical nth order polynomial and E is the passband ripple factor. The transfer function is normalized in frequency as we shall 118
Journal ofThe
Franklin
Institute
Nomographs and Filters TABLE
II
Functional scales for Fig. 5
W
f(w)= w-7
Iy,= (w - 7)/3
7
10
0
3
0
1
li w
I
IO
7
1 -
-
s/9
FIG. 5. Nomograph for u+u=w.
Vol.302,No. 2,August1976
119
.T. N. Hallberg and Claude S. Lindquist discuss in a moment. The maximum inband gain is normalized to unity or 0 dB. The typical magnitude response was shown in Fig. l(a). The nomograph of Fig. l(b) related the filter order n, the passband and stopband attenuation MP and M,, respectively, and the normalized stopband frequency R. For convenience, we re-express (24) as IM(jw)l’ Frequency
w is normalized
= 1 + ease.
(25)
as n=
w/w,
(26)
so in Fig. l(a) MP = IM(jI)l
(27)
M, = IM(jsZ)l. Substituting
(27a) into (25) shows that M; = 1 + &S:(l)
Therefore,
the frequency
normalization s:(l)
so that the ripple
parameter
= 1 + E’.
condition
(28)
is that
= 1
(29)
equals E = J(M2,-
1).
This equation shows that when Mp is chosen, determined. Substituting (30) into (25) yields IM(jw)l’=
1 +(Mg-
Substituting in the M, condition of (27b), Mp, M,, n, and fl all together as
(30) the passband
ripple
factor
1)$(w). we obtain
E is
(31) the equation
that relates
M: = 1 + (MS- l)S:(R).
(32)
This is the equation from which we will make our nomographs. It should be emphasized here that Mp and M, are numeric values and not in dB. We have introduced the fundamental ideas on construction of a nomograph from an equation with three variables. However, this equation has four variables (M,, M,, n, a). Fortunately after some study, it becomes apparent that the above equation leads to a special case of a four variable nomograph. nomographs” (14), “combiThese nomographs are referred to as “collineation nation nomographs” (15), and “grid nomographs” (8). We like to think of them as combination nomographs. To obtain them, Otto’s procedure (14) will now be described. Consider the equation (33)
f1(u) +f*(u) = f4(w, 0.
Then if the function f4(w, t) is neither a sum nor a difference of two functions each of which depends on one variable, we can replace this equation by two 120
Journal of The
Franklin
Institute
Nomographs and Filters equations where fI(U)
+f2(fJ)
Y =fJw,
= Y
(34)
0
For (34b), we must draw a family of curves in the Cartesian system with coordinate axis (7, t) or (7, w). We choose the axes y and w. The family of (y, w) curves are then functions of t. We then combine this drawing with the nomograph of (34a), composed of three scales. Note the y scale is identical with the previous y scale of (34b). This results in the final form of the nomograph for (33) as shown in Fig. 6. All that remains now is to express (32) in the form of (33). However (33) involves a product. A very simple technique, which is widely used in nomography to separate variables, is to take the logarithm of an equation. Rearranging (32) as M: - 1 = (M2,- l)S’,@)
(35)
and taking the log (base 10) yields log (MS - 1) = log @f;-
1) + log s’,(n)
(36)
log (IL45- 1) -log @I;-
1) = log s”,(n).
(37)
or Now with the minor exception of a sign (it will be taken care of later), this equation has the same form as (33). Therefore the nomograph will look like Fig. 6. Following the procedure of Otto, we will form the two equations of (34) and introduce y. Re-expressing (37) as log(i@-l)-log(M%-l)=y
Y
(38)
_. I I I
FIG.
Vol. 302, No. 2, August
1976
6.
Combination nomograph. 121
J. N. Hallberg and Claude S. Lindquist
then (39) Now consider (38) and (39). Notice that of the two, only (39) will change when different transfer functions are considered. This demonstrates that a nomograph from (38) can be used for all of the filters whose gain can be written in the form of (24). Equation (39) will be used to form the graph portion of the combination nomograph of Fig. 6. When (38) was derived, M, and MP were numeric values and not in dB as we would like. To convert them to dB, we make use of the following substitutions. Since M s cm= 20 log M,
(40)
or M, = ~O~V&~
(41)
then squaring this gives M: = 10”.de/lo. Expressing
(42)
Mp in the same form gives M’, = 10”~d~‘lo.
Using these last two equations
(43)
in (38) yields
( log (lO”JO-
I)-log
(lO+LJ’O-
1) = y 1
(44)
We can now use the two boxed equations, (39) and (44), to construct the combination nomograph for the various classical filters. From this point on, we will be using M,,, and M, dB. For notational simplicity, we will write these as Mp and M, from now on. First consider (44). We see that when Mp and M, are greater than 10 dB, the “1” can be dropped out of both log expressions. Eliminating the “1” in (44) reduces it to
M, -_Ap= 10
M 10
y,
Mp, M,>>lOdB.
The validity of this can be demonstrated
by simple evaluation:
1
log (lo-
1) = 0.9542
log (10’) = 2
log (lo*-
1) = 1.9956
log(103)=3
log(103-1)=2.99956.
log(lO)=
(45)
(46)
The error is less than 5% for MalOdB and less than 0.5% for Ms20dB. We see this is simply u + 2,= w and is the same equation as the example. Since we have not introduced scale moduli, the basic determinant for the filter 122
Journal of The Franklin
Institute
Nomographs and Filters
nomograph
from (8) and (20) (with m, = m, = 1) is 0 1 $
From the basic determinant,
M,/lO
1 1 =o. 1
Y MJ20
(47)
we see that x1=0
y1 = M&O
x2 = 1 (unit)
y2 = y
x3 = 4 (unit)
y3 = MJlO
(48)
Thus, the scale supports (i.e. axis Mp, MS, and y) can be drawn on Fig. 7 carefully observing that x3 is exactly half-way between x1 and x2 where x2 is arbitrary. Here x2 has been chosen to accommodate the a-axis to be discussed in a moment. Once the scale supports are laid down, we can mark off the scales. It is necessary to specify ranges for Mp and M.. After some deliberation, it was decided to use MP from 0.01 to 50 dB and MS from 0.1 to 150 dB. From (43, the y corresponding to the maximum Mp and MS is 10. Using cm graph
50-
150-
45-
140-
40-
13c--
35-
120-
30-
IIO-
25-
IOO-
20-
7
_‘O-F
go%,_
%5-
6_
70-
50l0,5O.I0.05oa-
6
&o-
40-
5 :
3020IO5 IO I-
31 2
3
4
5
6
769
10
R
FIG. 7. Finished nomograph
scales.
123
J. N. Hallberg and Claude S. Lindquist
paper, we arbitrarily scale the y-axis from 0 to 12 with 2 cm between major points. We also arbitrarily position the maximum MP = 50 dB and MS = 150 dB on a horizontal line with y = 10. This sets the rest of the scales as far as positioning goes (7). To produce this alignment, the equations of (48) then become x1=0 yr=M,/10+5 x2 = 1 (unit)
yz = y
x3 = 4 (unit)
y3 = MS/20 + 2.5
(49)
where y1 and y3 have been translated. The scale moduli for the three scales M,/lO, MJ20, and y are 1, 1, and 1, respectively. As long as Mp and M, are both larger than 20 dB, both scales are easily draw since (45) is valid (within 0.5%). Since the scale modulus for MS/20 is 1, the scale from 20 to 150 dB is drawn with 1 cm between major ,points (integer multiples of 10 dB). By the same method, we draw the Mr,/10 scale from 20 to 50 dB with 2 cm between major points (integer multiples of 10 dB). To check several points, notice that data points on our nomograph lie on horizontal lines (i.e. y1 = y2 = y3 in (49)) whenever the gains satisfy M,=2Mp+50,
M,>20dB
(50)
in which case M,>20dB
y=Mp/10+5=M,/20+2.5,
Also notice that whenever
M >gOdB s
the gains are equal, then from (44) -r=O
This condition must be satisfied forms another convenience check To obtain the scales for 20 dB for which the nomograph is being
for
M,=M,
(52)
by any nomograph for any (M,, MS) which point. and below for Mp and MS, we return to (44) constructed. The basic determinant of (44) is
0 log(lo”~lo-l) 1 Y 1 ~log(lo”“o-l) which reduces become
(51)
1 1 =o 1
(53)
to (47) for Mp, M, > 20 dB. Then the nomograph x1=0
yr=log(lO”~‘o-
x2= 1 (unit)
y2= y
x3=$(unit)
y3=ilog(10MJ10-1)+2.5
equations
1)+5 (54)
by analogy with (49). A Fortran program was written to obtain various (yr, y2, yJ) combinations from (54). These values allowed us to complete the Mp and M, scales. They were used for Mp, M, < 40 dB. This also verified that our assumptions leading to (45) were valid.
124
Journal of The Franklin
Institute
Nomographs and Filters
It is interesting to observe the behavior of yl and yz for MP, A4.<<10 dB. Using the Maclaurin series for ax as aX=l+xlna+
(x In a)’ 21 +***
where a = 10 and x = M/10, then for xc< 1, log (a” - 1) = log (x In a) = log x + log (In a)).
(56)
Since a = 10 and log (In 10) = 0.362, then from (54) Yl=log
&+ 1o
5.362 M,,ibf,<<
Yz=Y
10dB.
(57)
Therefore, y1 and y3 vary logarithmically for Mp and M, less than 1 dB. Since the scale modulus for log (&/lo) is 1, the Mp scale below 1 dB is drawn with 2 cm between major points (integer multiples of log (M&O)). Similarly, the A4, scale below 1 dB is drawn with 1 cm between major points (integer multiples of log (k&/10)). We see from (57) that y1 = y3 = 0 when Mp = i%f,= 10-4.362. We have completed our nomograph as far as the Mp, M,, and y scales are concerned. What remains is to draw a scale on the y support and to plot unique curves in the resulting grid. Recalling from (39) that y = log s:(Q)
(58)
s;(a) = @(“--m), R + a,
(59)
and observing that
for every rational filter having a gain of (23), then y=2(n-m)logO,
Q+w.
(60)
Thus, to produce asymptotically linear curves in the (y, a)-plane, we choose f-I to be a logarithmically-scaled axis. From experience, one cycle provides an adequate range for Q. This completes the construction of the nomograph in Fig. 7. Now we are ready to review the various classical filters and to plot their S,(Q) from (39).
IV. Classical Filter Polynomials Classical filters are filters whose magnitude response is related to a classical polynomial. They can be derived from a basic polynomial f of second degree where (16), (17) f(x) = (b-x)(x Vol. 302, No. 2, August 1976
- a)
61)
125
J. N. Hallberg and Claude S. Lindquist TABLE
III
Classical polynomials
(16)
General Jacobi polynomials f(X)=(b-x)(x-u)
P(x)=(b-x)‘(x-a)@
f(X)=(l-x)(x+1) =lPxz
simplification to the interval (-1, +1) b=+l I a=-I, Jacobi polynomials p(x)=(l-x)“(l+x)P simplification I a=@=h Ultraspherical polynomials p(i)=(l-x2)” simplification
f(X)=l-x’
by using the different weighting functions polynomials can be expressed in the form
S,(x) = ~
l
P(X)Kl
p listed
a>-1 p>-1 la\<00 lbl
cr>--1 p>-1 a=-1 b=+l A>-1
in Table
III.
All classical
(62)
~'"'[pW"Wl
where DC”) represents the nth derivative of the expression in brackets and K,, is which can be a constant which depends on n. The classical polynomials generated are listed in Table III. The most general polynomials are the hypergeometric polynomials F(a, b, c; x). With a, p > -1 and restricting the interfal (a, b) to (-1, l), the Jacobi polynomials result. If (Y= p, then these polynomials simplify into the ultraspherical or Gegenbauer polynomials. Since Jacobi polynomials satisfy p’,“.@‘(-w) = (_l)“p’,p’“‘(w)
(63)
and any polynomial P’, used in the gain expression (24) must be even, then (Y= /3. Thus, ultraspherical polynomials must be used. In order that F, have its monotonically for 1w I> 1, parameter zeros within 1wI < 1 and IF,,1 increase (Y> -1. (Y can therefore assume any value in the range -l Johnson and Johnson in that the other filter types 126
(64)
in Table IV. (Y= -0.5 produce the well-known Butterworth and filters, respectively. (Y= 0 produces the Legendre produces the modified n,. . . (any positive integer) introduced by Ku and Drubin in 1962 (18).Using - 1 results in the ultraspherical filters introduced by 1966 (19). They are credited for having pointed out were special cases of the ultraspherical filter. Journal ofTheFranklin
Institute
Nomographs and Filters TABLE IV. Classical filters based upon the ultraspherical polynomialwhich is based upon the moregeneral hypergeometricpolynomial
-4
Ultrospherlcol
Common
(cc)
Hypergeometrlc
nome
a3
Butterworth
IO
10th
M.A.L.
polynomials 3
F(U,b,CjX)
i
2 I.5
Jacob1 polynomials
n
PIUSO
) Lx) t 0.5
3rd
M.A.L.
2nd
M.A.L.
l-5
Ultraspherical
1st
M.A.L.
2nd
Cheybshev
lI Ultraspherical
0
‘-
Legendre
-0.25
--
-0.25
polynomials F,O
-05
--
1st Cheybyshev
-0.75
--
-0.75
-0-9
--
-0.9
--
-0.95
--
-0.9999
-0.95 - 0.9999
V. Ultraspherical
G(w) are formed
functions =
Ultraspherical Ultraspherica
I
Ultraspherical
(58). To find c(w),
given
by (24) where
F3w)12.
(65)
wxw)12
we use the recursion
(2a+n)F~(~)=x(2a+2n-l)F~_,(x)-(n-l)~-~(~) and
- Rwple)
by plotting Y = log
from
(Equal
Ultraspherical
Filters
Ultraspherical filters have magnitude
The nomographs
Ultraspherxal
(66) formula
n22
(67)
(19) F;;(x)= 1 C(x)
= x.
(68)
Since FE(l) = 1, (65) satisfies the frequency normalization condition of (29). A Fortran computer program was written to generate the ultraspherical polynomials (to orders 20) using (67) and (68), and y of (66). Nomographs were generated for the (Y values listed in Table IV and are shown in the Appendix in Figs. Al-A14. They are labelled with their common name and the value of alpha (CY)following it in parenthesis.
Vol.302,No.2,August 1976
127
J. IV. Hallberg and Claude S. Lindquist
Butterworth filters are the class of all-pole maximally-flat magnitude filters. They have no passband ripple and a magnitude function of
IHWI” =
1
+
(MFM)
e:w2”
where s;(w)= Since (S,(l)j=
W2”.
(70)
1, then from (58) y = log wzn = 2n log w.
(71)
These are straight lines of slope 2n so the Butter-worth filter nomograph is easily plotted using a straight-edge in Fig. Al. Using (Y= 10’ in (66) yielded the same results which were used to verify the ultraspherical filter program. Filter orders through 25 are shown. The modified associated Legendre (MAL) filters result for positive integer values of (Y.They use modified associated Legendre polynomials Pi”” which are obtained by differentiating an (n + m)-degree Legendre polynomial P,,+,,,m times where P’,“‘(x) = $
P,+,(x).
(72)
Excluding elliptic filters, they were the first 3-parameter filters where (n, E, a) could be selected. The lOth, 3rd and 2nd M.A.L. filters have the nomographs shown in Figs. A2, A3 and A4, respectively. Ultraspherical filters result for noninteger values of (Y> -1. The 1.5 ultraspherical filter has the nomograph of Fig. A5. The 1st M.A.L. filter has the nomograph of Fig. A6. 01 = 0.5 results in a Chebyshev filter of the second kind. This is not the common equal-ripple or Chebyshev filter (of the first kind) where (Y= -0.5. They have little in common except (Y’Sof opposite sign. They use Chebyshev polynomials of the second kind V,,(w). The nomograph is shown in Fig. A7. The Legendre jilters have (Y= 0. They use the Legendre polynomials P,,(w). Unfortunately, this filter is widely confused with the Papoulis or monotonic-L filter to be discussed later (20). The nomograph is shown in Fig. A8. The -0.25 ultraspherical filter has the nomograph shown in Fig. A9. This filter was discussed by Petrela and Budak (21). The Chebysheu jilter (of the first kind) or equal-ripple filter has (Y= -0.5. It uses Chebyshev polynomials of the first kind T,,(w). The nomograph is shown in Fig. AlO. The -0.75, -0.90, -0.95 and -0.9999 ultraspherical filters have nomographs shown in Figs. All-A14. These are not very practical because of the large ripple as shall now be discussed. The ultraspherical filters of this section (a = 00+ -1) have been ordered in terms of increasing bandedge selectivity, i.e., -d IHJ/dw I,,,=i. Several magnitude responses are shown in Fig. 8 where n = 10 and E = 1 (21). The filters 128
Journal
of The Franklin
Institute
Nomographs and Filters
Frequncy,
rod/see.
w
FIG. 8. Magnitude response of several 10th order ultraspherical
filters having E = 1 (21).
have 3 dB frequencies of unity since E = 1. When a B-0.5, the inband ripple increases as the passband edge is approached. For -1
Magnitude Filters
There are other classical filters which are not derived directly from ultraspherical filters and which must be considered separately. These include the Papoulis, Halpern, Bessel, Gaussian, and synchronously-tuned filters which we shall now discuss. The Papoulis filter, sometimes called the monotonic-l or L-filter, was introduced by Papoulis in 1957 (20). As mentioned before, it is often mistakenly called a Legendre filter. It combines the desirable features of the Chebyshev and Butterworth filters by having (1) monotonic passband response, and (2) the maximum bandedge selectivity possible. The polynomials L,(w2) of the Papoulis filter are related to and derived from the Legendre polynomials (22)-(24). Using L1-L8 from (22), its nomograph was plotted in Fig. A15. It is interesting to observe that the Papoulis filter nomograph is almost identical to that of the 1st M.A.L. Thus, their stopband responses are very similar (of course, their passband responses are quite different). Halpern introduced another monotonic response filter in 1969 (25). It is related to the Papoulis filter in the following way: Papoulis maximized the rate of change of IS,,(w)] at w = 1, (djS,(w)]/dwl,=l) under the constraint of response. Halpern maximized the asymptotic growth of monotonic IS,,(w)l ((S,(w)l/w”) under the constraint of monotonic response. The polynomials H,,(w’) of the Halpern filter are related to and derived from the Jacobi Vol.302,No.2,August 1976
129
J. N. Hallberg and Claude S. Lindquist
polynomials. Using HI-H7 in (25), the nomograph was plotted in Fig. A16. It can be shown that the improvement in stopband rejection of the Halpern filter over that of the Papoulis filter is about OSJn. Since this is only about 1.7 or 4.7 dB for n = 10, the two nomographs are about the same except for large n. VII. Linear Phase/Constant
Delay Filters
The filters which have been considered so far have approximated magnitude response of the ideal lowpass filter having gain
the
H(jw) = eviTow, 1w I< 1
(73)
= 0,
Iwl’l
which equals unity for 1w ( < 1 and zero elsewhere. Now we shall consider filters which approximate the phase and delay response of the ideal filter where
Iwl
arg H = - WT~,
= arbitrary,
Iw I > 1
(74)
d arg H r=
Iwl
-dw=rO~
= arbitrary,
Iw I > 1.
These are called linear phase or constant delay filters. The Bessel or Thomson filter is the class of all-pole maximally-flat delay (MFD) or linear phase filter (26), (27). They approximate delay in the same way as Butterworth filters approximate magnitude. They were introduced by Thomson in 1949. Their transfer functions equal
H,(s) = where the polynomials by (28)
_ Qn(O)
a0 ao+aIs+.
. .+a,s”
(75)
Q,(s)
Q,(s) are obtained from the Bessel polynomials
Q,(s) = s”J%(l/s).
B,(s) (76)
It is important to note that this is different than say, for example, the Chebyshev or Legendre filters which use their respective polynomials directly. Thomson gives the squared modulus polynomial as
Q(lljw)Q(-lljw)
= f B,~(w*)~
(77)
k=O
where (n + k)! (2k)!
(78)
Bnk=(n-k)![k!2k]2.
Wood gives the same result and the useful recursion formula (29) & 130
= (2n - 1)(2k - l)Bn-r,k-i
+ B,,-_z,k
(79) Journal
of The Franklin
Institute
Nomographs
and Filters
where
Bno= 1 B
nl
(80)
=n(n+l> 2
.
A Fortran computer program was written to generate the Bessel filter polynomials to order 20. Using these polynomials, the squared-magnitude function can be written as
bo
IfL(j
= bo+blW2+.. .+bnW2n 1 1+;w2+.. 0
Therefore,
from
(81) .++w2" 0
(24), then
s’,(w)=+ w*+.. .+; W2n. 0 1
(82)
Frequency normalization by w3 dB is required to force (S,(l)/ = 1. w3 dn was found using iterative techniques. The nomographs for the Bessel filter is shown in Fig. A17. For n large and w small, the filter is Gaussian (28) as we shall verify. The Gaussian filter was introduced by Dishal in 1959 (30). The magnitude response of the ideal Gaussian filter equals IH(
=
l/IM(jw)l
where IH( =$. Expanding /HI’= truncating at the nth term gives
=
e-“~347w2= e-aw2’2
exp (-aw2)
(83)
in a Maclaurin
IM(jw)l’ = eaw2g l+2(aw)2+g(aw)4+.
. .+$
(aWyn
series
and
(84)
which is the squared-magnitude response of an nth order Gaussian filter. A Fortran program was written to generate these polynomials to order 20. Since the magnitude response has the form of (81), then S”, is found by using (82). Frequency normalization by wj dB is required to force (S,( l)( = 1. wsdB was solved for using iterative techniques. The nomograph for the Gaussian filter is shown in Fig. A18. As n + m, the response becomes truly Gaussian. For large w, then
S2(w) = e2(o.347wz),1w 1>>1 Vol. 302, No. 2, August 1976
(85) 131
J. N. Hallberg and Claude S. Lindquist
and y=10gs2=0.3w2,
}w(>>l.
(86)
This is the limiting curve as n + 00 in the nomograph. Thus, as with the Bessel filter, there is a maximum rejection which can be obtained independent of order. Note that the Bessel nomograph approaches the Gaussian nomograph for large n and w = 1 (this is the “peculiar” cross-over effect). Synchronously-tuned filters have the simplest transfer function of all filter types. They consist of n identical cascaded filter stages all having the same simple pole po. They have a squared-magnitude function of
IHntiw)l*= ,l+~~~~. To maintain unity 3 dB bandwidth,
the pole p. must be located as
J(n)
Po=(2”“-l)-f+__
J(ln 2) a Using the binomial theorem
(87)
W
to expand IMn(iw)12 as
where the coefficients of (w/pa)* are the well-known binomial coefficients. since (90) a Fortran program was written to calculate y for orders to 20. The resulting nomograph is shown in Fig. A19. Noting that for large n, the magnitude response can be expressed as IHO.w)l=!?_[l+$]-“‘*
(91)
where 0 = J(ln 2) w, then making the change of variable v= n/Cl, we obtain
(92) which is Gaussian from (83). Thus, the synchronously-tuned filter also has the same limiting y as the Gaussian filter. It may be questioned why we treat the 132
Journal of The Franklin Institute
Nomographs and Filters synchronously-tuned filter as a constant delay filter. The reason is that its selectivity is low, its delay response is fairly flat, and its step response is monotonic (2) which is similar to the other constant delay types. VIII. Usefulness of Filter Nomographs We have reviewed a variety of classical filters and presented a collection of nineteen nomographs which describe their magnitude responses. We can easily group these filters in two main categories as: (1) High selectivity filters (2) Low selectivity filters based on the shape of the y = log S’, curves. Using the linear curves given by (60) as a reference (e.g. the Butterworth filter curves), curves convex above the reference are highly selective; those convex below the reference (or concave) have low selectivity. The ultraspherical filters are highly selective (but large ripple) for (r = -1. They become decreasingly selective for increasing cz + 00. The Papoulis and Halpern filters had selectivities close to those for 1.0 ultraspherical filters. The constant delay filters; Bessel, Gaussian, and synchronously-tuned filters, were very selective (increasingly so in this order). This gives a much more general interpretation of selectivity than has been available in the past. It also allows filters to be easily compared. The nomographs make the determination of filter order trivial. For example, suppose that the filter orders must be determined to meet the response of Fig. 9. Expressing the parameters in the form of Fig. l(a) as M,=ldB M,=40dB
at
R=l.S
(93)
we enter this data on the nomographs in the Appendix as shown in Fig. l(b) (Y = 4.6 and fi = 1.5). Then the minimum filter orders are listed in Table V. We see that none of the constant delay filters can be used. We can use a 14th order Butterworth,,8th order Papoulis, 8th order Legendre, or 7th order Chebyshev
FIG.
9. Example of required response of a lowpass filter.
Vol. 302, No. 2. August 1976
133
J. N. Hallberg and Claude S. Lindquist TABLE V Minimum order filters for filter response of Fig. 9
Filter type Chebyshev (-0.5) -0.25 ultraspherical Legendre (0) Chebyshev (0.5) Halpern Papoulis 1st M.A.L. (1) 1.5 ultraspherical 2nd M.A.L. (2) 3rd M.A.L. (3) 10th M.A.L. (10) Butterworth (m) Bessel Gaussian Synchronously-tuned
(equal-ripple) filter. order via equations nomographs.
Minimum order n 7 7 8 8 8 8 8 9 9 10 11 14 Impossible Impossible Impossible
Any engineer who has wrestled with determining filter will immediately recognize the tremendous value of
IX. Comparison with Earlier Work We mentioned earlier that (1)and (2) contain certain inaccuracies. These can produce errors when determining order which result in filter designs which do not exhibit the required stopband rejection. This is of course puzzling when the inaccuracies are not recognized. Therefore, it is important to identify the problems. Kawakami (1)used analogous equations for y [see Eqs. (39) and (44)]. His plots for y = log S’,(n) = r(n) appear to be correct. However, the y-axis has been shifted slightly (to the right for Butterworth and the left for Chebyshev). This may be verified by checking y = 0 for several Mp = M, combinations [see Eq. (52)]. These errors appear to be compounded in (2). We see that it is vitally important to insure the exact geometric relation required by (20) for scale spacing, (53) for marking of the scales, and (39) for plotting y vs. 0. These observations prompted us to design our nomograph of Fig. 7 with maximum symmetry. We were careful to choose scale factors m, and m, of unity so (x1, x2, xg)= (0,1, $) in (20). References (1) and (2) use about (0 1,0.56). With our choice of scale factors, (u, U, w) = (M,/lO, y, M,/20) (assuming Mp, M, >>10 dB) all had unity scale factors which insured equal spacing between major points of i’$/lO (10 dB steps), MJ20 (20 dB steps), and y (unit steps). By translating (u, U, w) so that (50 dB, 10, 150 dB) all lie on the
134
Ioumal of The Franklrn
Institute
Nomographs same
choice
and Filters
horizontal line, the scales were easily marked off and completed. of scale factors in (1)and (2) did not yield this convenience.
The
X. Conclusions Filter nomographs provide a graphical means for determining filter order and also tradeoffs in maximum passband attenuation MP, minimum stopband rejection MS, normalized stopband frequency &I, and order n. They also allow the stopband responses of various filters to be compared. Thus, they provide maximum information with a minimum of time invested. The basic nomograph axes are easy to construct once basic nomograph theory is understood. The same axes can be used for any filter. Only the y = -y(a) curves must be modified to match the standard response functions being used. The equations which describe these curves are both complicated and long so they require computer evaluation. However, once the nomograph has been obtained, it can be used for all future design. This paper has presented the nomographs of 19 classical filters of which 17 nomographs are new. The other two nomographs replace the earlier nomographs which contain inaccuracies. References (1) M. Kawakami, “Nomographs for Butterworth and Chebyshev filters,” IEEE Trans. Circuit Theory, Vol. CT-lo, pp. 288-289, June, 1963. (2) A. I. Zverev, “Handbook of Filter Synthesis,” Wiley, New York, 1967. (3) E. Christian and E. Eisenmann, “Filter Design Tables and Graphs,” Wiley, New York, 1966. (4) J. N. Hallberg, “Filters and nomographs,” M.S.E.E. Directed Research, California State Univ., Long Beach, Ca., Jan., 1974. (5) A. Budak, “Passive and Active Network Analysis and Synthesis,” Houghton Mifflin, Boston, 1974. (6) M. S. d’ocagne, “Traite de Nomographie,” 2nd ed., Paris, 1921. (7) L. Johnson, “Nomography and Empirical Equations,” Wiley, New York, pp. 19-33, 1952. (8) H. J. Allcock, “The Nomogram,” 5th ed., Pitman, London, 1963. (9) L. I. Epstein, “Nomography,” Interscience, New York, 1958. (10) J. W. Richards, “Introduction to Graphs and Nomograms,” Heywood Books, London, pp. 98-110, 1966. (11)A. S. Levens, “Nomography,” Wiley, New York, 1959. (12) R. D. Douglass, “Elements of Nomography,” McGraw Hill, New York, 1947. (13) J. H. Fasal, “Nomography,” Ungar, New York, 1968. (14) E. Otto, “Nomography,” MacMillan, New York, 1963. (15) D. Davis, “Nomography and Empirical Equations,” Reinhold, New York, 1962. (16) J. Vlach, “Computerized Approximation and Synthesis of Linear Networks,” Chap. 6, Wiley, New York, 1969. (17) M. Abramowitz and I. A. Stegan (Eds.), “Handbook of Mathematical Functions,” Natl. Bur. of Standards, Vol. 5.5, Sect. 22, 1972. (18) Y. H. Ku and M. Drubin, “Network synthesis using Legendre and Hermite polynomials,” J. Franklin Inst., Vol. 273, pp. 138-157, Feb., 1962. filters using ultraspherical (19) D. E. Johnson and J. R. Johnson, “Low-pass mials,” IEEE Trans. Circuit Theory, Vol. CT-13, pp. 364-369, Dec.,
Vol. 302, No. 2, August 1976
polyno1966.
J. N. Hallberg
and Claude S. Lindquist
(20) A. Papouiis, “On the approximation problem in filter design,” IRE Natl. Cow. Rec.. part 2, pp. 175-185. 1957. (21) D. M. Petrela and A. Budak, “Pole locations for ultraspherical filters,” IEEE Tram Circuit Theory, Vol. CT-17, pp. 668-670, Nov., 1970. (22) A. Papoulis, “Optimum filters with monotonic response,” Proc. IRE, Vol. 46, pp. 606-609, March, 1958. (23) A. Papoulis, “On monotonic response filters,” Proc. IRE, Vol. 47, pp. 331-333, Feb., 1959. (24) M. Fukada, “Optimum filters of even orders with monotonic response,” IRE Trans. Circuit Theory, Vol. Cl’-6, pp. 277-281, Sept., 1959. (25) P. H. Halpern, “Optimum monotonic low-pass filters,” IEEE Trans. Circuit Theory, Vol. (X-16, pp. 240-241, May, 1969. (26) W. E. Thomson, “Maximally-flat delay networks,” IRE Trans. Circuit Theory, Vol. CT-16, p. 235, June, 1959. (27) W. E. Thomson, “Delay networks having maximally flat frequency characteristics,” Proc. IEE, pt. 3, Vol. 96, pp. 487-490, Nov., 1949. (28) L. Starch, “Synthesis of constant time delay ladder networks using Bessel polynomials.” Proc. IRE, Vol. 42, pp. 1666-1675, NOV., 1954. (29) I. E. Wood, “Note on maximally flat delay networks,” IRE Trans. Circuit Theory, Vol. CT-5, pp. 363-364, Dec., 1958. (30) M. Dishal, “Gaussian response filter design,” Electrical Commun. Vol. 36, No. 1, pp. 3-26, 1959.
50-
I50’-
45-
l40-
40-
130-
35-
IZO-
30-
IIO-
25-
lOO---
20,” L
so-
15-g
EO-
IO-
TO-
$
5_$6050 -I-
40-
0.5-
30-
O.I0.05 -
20
00--
IO5, 0.1 -
FIG. Al. Nomograph for Butterworth
136
(m) filters. Journal of The Franklin Institute
Nomographs and Filters
Vol. 302, No. 2, August 1976
137
J. N. Hallberg and Claude S. Lindquist
138
Journal of The Franklin
Institute
FIG. A6.
Nomograph for first modified Legendre (1) filters.
0.1-
IO5I-
20-
30-
associated
FIG.
A7.
o.o+
0.1 0.05 i
o-5
Nomograph
0.1-
I-
IO5
20-
30-
5040-
_?Os*"sO-
40-
5
," 80--
go-
15
20
* '0 s"
:
$50-
.a-
70-
%80-
IIOIOO-
25
120-
35 3
140130-
40
15045
for Chebyshev second kind.
(0.5)
filters-
3 9 II
_L_
I
/
J. N. Hallberg and Claude S. Lindquist
140
Journal of The Franklin
Imtitutc
Nomographs and Filters
J. IV. Hallberg and Claude S. Lindquist
142
Journal of The Franklin Institute
Nomographs and Filters
Vol. 302, No. 2, August 1976
143
J. N. Hallberg and Claude S. Lindquist
I I I 000000000 LOZ’OEZQrnW
144
I
I I
I 11
1 0Ij
0
~(0u-JbmN-
EP ’ %/
0
0
110
IIll om
-
-
I
6
Journal
of The Franklin
d !I
Institute
&jl
m -J
15-
-
1
I
2
3
a
4
5
678910
FIG. Al% Nomograph for Gaussian filters.
O.ly
5-
20,O_
O.IC 0.05-
0.01
30-
4O-
g5O-
-60
0.5-
I-
5-
?O-
go-
so-
20-
‘O-m,
IIO-
IOO-
30-
35-
25-
130-
120-
40--
150-
140-
50-
45-
iE IO 5
20
30
40
2
3
4
5
678910
FIG. A19. Nomograph for synchronously-tuned
0.01
o- I 0.05
0.5
filters.
Volume
302,
Number
TO SCIENCE
AND
Institute THE MECHANIC
ARTS
August
2
1976
Nomographs and Filters by JOE N. HALLBERG
Interstate Electronics Corporation, 707 E. Vermont, Anaheim, California
and
CLAUDE
s. LINDQUIST
Department of Electrical Engineering, California State University, Long Beach,
California
AEISTRACKThis paper presents the nomographs for additional classical filters, including ultraspherical, Legendre, modified associated Legendre, Papoulis, Halpern, Bessel, Gaussian, and synchronously-tuned. It also identifies inaccuracies in the earlier nomographs. The basic theory of nomographs and their utilization in developing filter nomographs is presented.
I. Introduction Nomographs have been used extensively for a half century in the sciences for graphically representing mathematical relationships. In 1963, Kawakami applied nomography to the area of classical lowpass filter response (1).He introduced three nomographs which described Butterworth, Chebyshev, and elliptic filters. These nomographs depicted the interrelationship between the maximum passband attenuation Mp, the minimum stopband rejection MS, the normalized stopband frequency R, and the filter order n as shown in Fig. 1. Given any three of these variables, the fourth could be easily determined by inspection. As an example, if MP = 1 dB, MS = 40 dB, and Sz = 1.5, then the order n of the filter will be obtained as the figure demonstrates. This capability is invaluable in filter design. Later in 1967, similar nomographs were published by Zverev (2) and have become widely used by industry in passive and active filter design [for an alternative form, see (3)]. Unfortunately, with the myriad of other filter types commercially available and in use (e.g. Papoulis, Bessel, etc.), no additional nomographs have emerged. Not only has this hindered rapid design, but more importantly, it has hampered easy comparison of the various filter types. This, coupled with the fact that the nomographs of (1)and
111
J. N. Hallberg and Claude S. Lindquist
FIG. 1. (a) Typical
magnitude
response
of lowpass filter and (b) nomograph nitude response.
of mag-
(2) contain inaccuracies, has motivated the study of nomographs and filters presented in this paper (4). This paper addresses and discusses four major topics: the general theory of nomographs, application of nomography to filters, the classical filter responses which are widely known, and the nomographs which describe these responses. It will be shown that the theoretical basis of filter nomographs is rather simple. Yet most engineers find it surprising that so much filter information can be obtained from them. These nomographs provide truly marveilous design aids as we shall see. We will also present a design example, and identify inaccuracies in the earlier work. 112
Journal of The
Franklin
Institute
Nomographs and Filters 5-
10
5
/ /' /'
" u
I' '6
3p
"3
/' /' 1' /'
2L
4
2
2
4--
FIG. 2.
Nomograph
0/8
0 i
for u + 2)= w. Given u = 2 and u = 4, then w = 6.
We should also note the generality of this work. The nomographs to be constructed will describe the magnitude response of lowpass filters. By use of the Ibwpass-to-highpass, lowpass-to-bandpass, and lowpass-to-bandstop transformations (5), they can equally well be applied to these other classes of filters. II. General
Theory of Nomographs
Nomographs derive their name from the Greek words nomos (law) and graphein (to write). They are literally graphical representations of mathematical laws or relationships. Nomography, the study and use of nomographs, was introduced in 1891 by Lt. Col. M. d’ocagne, the French mathematician, in his “Traite De Nomographie” (67). Nomography quickly spread to England, Germany, and other parts of the world. The Germans refer to them as nomograms and the English call them nomographs. Occasionally, they are called alignment charts. In this paper, they will be referred to as nomographs. A nomograph is simply a graphical representation of equations in three or more variables. We will first describe a nomograph with three variables. Each variable is plotted on a line. Given any two, the third can be obtained. Drawing a straight line between the given two, the intersection on the third line gives the value of the third variable. An example of a nomograph is shown in Fig. 2. We shall now discuss the theoretical basis for nomographs. d’Ocagne based his development of nomographs upon projective geometry and the principle of duality, using parallel line coordinates (6). We will use the approaches of Allcock (8) and Epstein (9) that use determinants and Cartesian coordinates. The most fundamental principle of all nomographs is that any three points, say (xi, yr), (x2, y2), and (x3, y3), lie on the same straight line when the value of the determinant x1
Vol. 302, No. 2. AugustI!?76
Yl
1
x2
y2
1
x3
y3
1
(1)
113
J. N. Hallberg
and Claude
S. Lindquisl
xI FIG. 3.
x
5
x2
Three functions u, v, and w.
is zero (10). This can easily be shown to be true (9) by letting u, u), and w, denote the three functions on which points (x1, yi), (x2, yz), and (x3, y3) lie as shown in Fig. 3. Then let A, B, and C be the intersections of an arbitrary straight line with the u, 21,and w curves, respectively. Denote point (x2, yi) by D and (x3, y2) by E. Thus, AD = x2-x1 and DB = y2- yl. Similarly, EC= y3 - y2 and BE = x3 - x2_ Equating the hypotenuse slopes of triangles ABD and BCE, we see y3-y2 y2-=
Clearing
the fractions
(2)
x3-x2’
x2-x1
and rearranging,
we obtain (3)
~ly2+~2y3+~3yl-~ly3-~2yl-~3y2=0~
This may be written
in determinant
notation
as
Xl
y1
1
x2
y2
1 =o.
I x3
y3
1
(4)
I
Now, if xi and yi are functions of u only, x2 and y2 of v only, and xg and y3 of w, we can rewrite the determinant as &5(u)
f,(u)
I
g2(u)
f2(u)
1
g3(w)
f3(w)
1
=o.
(5)
This determinant is often called the basic determinant. If any equation can be written so it is possible to form the basic determinant, then it is possible to construct a nomgraph consisting of three curves in the (x, y)-plane which are graduated and represent the variables u, v, and w (8). 114
Journalof The
Franklin Institute
Nomographs and Filters
It is important to re-emphasize that the basic determinant of any nomograph must satisfy the following conditions: (a) The value must be zero. (b) Each row must contain one variable only. (c) The last element of each row must be positive and equal to unity. As we shall now discuss, the nomograph is formed from the basic determinant by plotting the points [gl(u)=xl, f~(u>=~11, [gdu) =x2, Mu)= ~21, and [&(W)= x3, f3(w) = Y31,for various values of U, u, and w. This produces the lines of the nomograph as shown in Fig. 2. The first step in designing a nomograph is the formation of the basic determinant. The general method of deriving the basic determinant from any formula containing three variables consists, essentially, in forming three simultaneous equations of the form Ax +By + C = 0. From these equations, the determinant
i
AI
6
G
A2
B2
C,
A3
B3
G
=0
(6)
I
may be derived directly by writing the coefficients of the equations in the order shown. Then this determinant is transformed into the basic nomographic form by taking advantage of any of the well-known determinant manipulations. We now illustrate the formation of the basic determinant with an example (11). We design the nomograph shown in Fig. 2 for the equation u+v=w
(7)
since this equation will be utilized later in filter design. First, we rewrite the equation as u+v-w=o.
(8)
Then we introduce the parametric equations x = u and y = 2). With these two equations and the one above, we have the required three where x-u=0 y-v=0
(9)
x+y-w=o. Substituting these equations into the determinant
I 01
01
-w -u -v
I
=o.
form of (6), we obtain
(IO)
It now remains to express this determinant as a basic determinant in the standard form of (5). This is accomplished by using basic determinant manipulations as will now be described. In the above determinant, column 1 is replaced by the sum of columns 1 and Vol. 302, No. 2, August 1976
115
J. N. Hallberg
and
Claude
S. Lindquist
2, and column
3 is multiplied
by -1,
i 21 The bottom
row is divided
yielding
w u2, I =o.
01
by 2; resulting 10
By interchanging
columns,
(11)
in u
1 I 1
1
z1
1
t
WI2
=o.
the final form is achieved
I
0
ul
1 1
v
2
w/2
(12) as
1 =o
(13)
1I
which is the basic determinant. Note that all the steps shown above were permissible because the value of the determinant was zero. The nomograph of Fig. 2 is constructed from this basic determinant. Comparing this basic determinant to (4), the equations are:
The nomograph w, and v at
xi = 0
y1=u
x2=1unit
y2=v
x3 = 4 unit
y3 =
is constructed
by first drawing
(14) w/2. the three
“scale supports”
for u,
xr=o x3=+
(I%
x2= 1. Then
the “functional
scales”
are drawn for u, w, and v. The functional scale are marked with the “value of the variable” u, w, or v and on which the graduations are laid off in proportion to the corresponding values of the “function of the variable” f(u), f(w), or f(v). The distances are laid off from an initial point of the scale, which are not necessarily the zero points. Although it is not necessary in this simple example, tables are often used to describe the functional scales. Defining u and v to vary from 0 to 5 (cm) and w to vary from 0 to 10 (cm), then f(u) and f(v) are listed in Table I(a) and f(w) in Table I(b). The functional scales are then drawn in Fig. 4. This completes the nomograph for u + 2) = w. When the function values, f(u), f( v ) , or f(w), do not lie in the proper range is used. The scale modulus m is a scale for plotting, a “scale modulus” multiplier which enables us to place the value of a function within a prescribed range (12), (13). Mathematically, the modulus m equals
(11)is one on which graduations
(16)
116
Journal of The Franklin Institute
Nomographs and Filters TABLE Functional
I
scales for Fig. 4
where x is the length, fHand fL are the values of the functions at the upper and lower limits of U, and (fH- fL) is the range of the function (13). This is one of the most useful equations in nomography. For example, consider the equation u + u = w once again. However, this time assume that u varies from 2 to 10 and that II varies from 5 to 15. If the lengths of the u and v are scales of 6 cm, then the scale moduli are
mu===;
6
3
6 m, =15_5=?.
(17)
3
Now let us see the effect these scale moduli have on the nomograph
FIG.~. Nomograph
for u+u=
w.
of Fig.
J. N. Hallberg and Claude S. Lindquist 4. Scaling u and v by scale moduli m,, and m, in (9) gives x - m,,u = 0 y-m,v=O
(18)
x/m, + y/m” - w = 0.
We can reform determinant
(10) as
-m,u 0 -r&v =o. 0 1 l/m, l/m, -w is easily reduced to the basic determinant
1
This determinant
0 1 mJ(m, + mv)
m,u r&u m,nkw/(mU + mu)
(19) form as
1 1 =o
(20)
1
For m, =a and m, = 2, then (20) equals (21) To draw the nomograph, we adjust the functions of u, v, and w so that the zero of measurement (i.e. y = 0) coincides with the first values of the functions (11). This usually requires simple translation of u, v, and w. Translating u, v, and w in (21), then 10 3(u-2)/4 11 1 3(v-5)/5 1 =o. (22) d (w-7)/3 1 We can now form a table of the functional scales of the nomograph as shown in Table II. From Table II and the x coordinates in the basic determinant (22), the nomograph is constructed in Fig. 5. The basic determinant (20) is the general determinant of the equation u + v = w. It reduces to (13) by letting m, = 1 and m, = 1. Now let us apply these results to filter analysis. III. Applying Nomography
to Filters
The transfer function H(s) of an nth order lowpass filter equals 1 N,(s) H,(s)=-=M”(S) D,(s)’ m< n where A4 denotes its reciprocal, and N and D are polynomials filters have squared magnitude responses of lH(jwV=
(23)
in s. Classical (24)
,Mfjw),2 = 1 + E21si,(w)
where S,(w) is related to a classical nth order polynomial and E is the passband ripple factor. The transfer function is normalized in frequency as we shall 118
Journal ofThe
Franklin
Institute
Nomographs and Filters TABLE
II
Functional scales for Fig. 5
W
f(w)= w-7
Iy,= (w - 7)/3
7
10
0
3
0
1
li w
I
IO
7
1 -
-
s/9
FIG. 5. Nomograph for u+u=w.
Vol.302,No. 2,August1976
119
.T. N. Hallberg and Claude S. Lindquist discuss in a moment. The maximum inband gain is normalized to unity or 0 dB. The typical magnitude response was shown in Fig. l(a). The nomograph of Fig. l(b) related the filter order n, the passband and stopband attenuation MP and M,, respectively, and the normalized stopband frequency R. For convenience, we re-express (24) as IM(jw)l’ Frequency
w is normalized
= 1 + ease.
(25)
as n=
w/w,
(26)
so in Fig. l(a) MP = IM(jI)l
(27)
M, = IM(jsZ)l. Substituting
(27a) into (25) shows that M; = 1 + &S:(l)
Therefore,
the frequency
normalization s:(l)
so that the ripple
parameter
= 1 + E’.
condition
(28)
is that
= 1
(29)
equals E = J(M2,-
1).
This equation shows that when Mp is chosen, determined. Substituting (30) into (25) yields IM(jw)l’=
1 +(Mg-
Substituting in the M, condition of (27b), Mp, M,, n, and fl all together as
(30) the passband
ripple
factor
1)$(w). we obtain
E is
(31) the equation
that relates
M: = 1 + (MS- l)S:(R).
(32)
This is the equation from which we will make our nomographs. It should be emphasized here that Mp and M, are numeric values and not in dB. We have introduced the fundamental ideas on construction of a nomograph from an equation with three variables. However, this equation has four variables (M,, M,, n, a). Fortunately after some study, it becomes apparent that the above equation leads to a special case of a four variable nomograph. nomographs” (14), “combiThese nomographs are referred to as “collineation nation nomographs” (15), and “grid nomographs” (8). We like to think of them as combination nomographs. To obtain them, Otto’s procedure (14) will now be described. Consider the equation (33)
f1(u) +f*(u) = f4(w, 0.
Then if the function f4(w, t) is neither a sum nor a difference of two functions each of which depends on one variable, we can replace this equation by two 120
Journal of The
Franklin
Institute
Nomographs and Filters equations where fI(U)
+f2(fJ)
Y =fJw,
= Y
(34)
0
For (34b), we must draw a family of curves in the Cartesian system with coordinate axis (7, t) or (7, w). We choose the axes y and w. The family of (y, w) curves are then functions of t. We then combine this drawing with the nomograph of (34a), composed of three scales. Note the y scale is identical with the previous y scale of (34b). This results in the final form of the nomograph for (33) as shown in Fig. 6. All that remains now is to express (32) in the form of (33). However (33) involves a product. A very simple technique, which is widely used in nomography to separate variables, is to take the logarithm of an equation. Rearranging (32) as M: - 1 = (M2,- l)S’,@)
(35)
and taking the log (base 10) yields log (MS - 1) = log @f;-
1) + log s’,(n)
(36)
log (IL45- 1) -log @I;-
1) = log s”,(n).
(37)
or Now with the minor exception of a sign (it will be taken care of later), this equation has the same form as (33). Therefore the nomograph will look like Fig. 6. Following the procedure of Otto, we will form the two equations of (34) and introduce y. Re-expressing (37) as log(i@-l)-log(M%-l)=y
Y
(38)
_. I I I
FIG.
Vol. 302, No. 2, August
1976
6.
Combination nomograph. 121
J. N. Hallberg and Claude S. Lindquist
then (39) Now consider (38) and (39). Notice that of the two, only (39) will change when different transfer functions are considered. This demonstrates that a nomograph from (38) can be used for all of the filters whose gain can be written in the form of (24). Equation (39) will be used to form the graph portion of the combination nomograph of Fig. 6. When (38) was derived, M, and MP were numeric values and not in dB as we would like. To convert them to dB, we make use of the following substitutions. Since M s cm= 20 log M,
(40)
or M, = ~O~V&~
(41)
then squaring this gives M: = 10”.de/lo. Expressing
(42)
Mp in the same form gives M’, = 10”~d~‘lo.
Using these last two equations
(43)
in (38) yields
( log (lO”JO-
I)-log
(lO+LJ’O-
1) = y 1
(44)
We can now use the two boxed equations, (39) and (44), to construct the combination nomograph for the various classical filters. From this point on, we will be using M,,, and M, dB. For notational simplicity, we will write these as Mp and M, from now on. First consider (44). We see that when Mp and M, are greater than 10 dB, the “1” can be dropped out of both log expressions. Eliminating the “1” in (44) reduces it to
M, -_Ap= 10
M 10
y,
Mp, M,>>lOdB.
The validity of this can be demonstrated
by simple evaluation:
1
log (lo-
1) = 0.9542
log (10’) = 2
log (lo*-
1) = 1.9956
log(103)=3
log(103-1)=2.99956.
log(lO)=
(45)
(46)
The error is less than 5% for MalOdB and less than 0.5% for Ms20dB. We see this is simply u + 2,= w and is the same equation as the example. Since we have not introduced scale moduli, the basic determinant for the filter 122
Journal of The Franklin
Institute
Nomographs and Filters
nomograph
from (8) and (20) (with m, = m, = 1) is 0 1 $
From the basic determinant,
M,/lO
1 1 =o. 1
Y MJ20
(47)
we see that x1=0
y1 = M&O
x2 = 1 (unit)
y2 = y
x3 = 4 (unit)
y3 = MJlO
(48)
Thus, the scale supports (i.e. axis Mp, MS, and y) can be drawn on Fig. 7 carefully observing that x3 is exactly half-way between x1 and x2 where x2 is arbitrary. Here x2 has been chosen to accommodate the a-axis to be discussed in a moment. Once the scale supports are laid down, we can mark off the scales. It is necessary to specify ranges for Mp and M.. After some deliberation, it was decided to use MP from 0.01 to 50 dB and MS from 0.1 to 150 dB. From (43, the y corresponding to the maximum Mp and MS is 10. Using cm graph
50-
150-
45-
140-
40-
13c--
35-
120-
30-
IIO-
25-
IOO-
20-
7
_‘O-F
go%,_
%5-
6_
70-
50l0,5O.I0.05oa-
6
&o-
40-
5 :
3020IO5 IO I-
31 2
3
4
5
6
769
10
R
FIG. 7. Finished nomograph
scales.
123
J. N. Hallberg and Claude S. Lindquist
paper, we arbitrarily scale the y-axis from 0 to 12 with 2 cm between major points. We also arbitrarily position the maximum MP = 50 dB and MS = 150 dB on a horizontal line with y = 10. This sets the rest of the scales as far as positioning goes (7). To produce this alignment, the equations of (48) then become x1=0 yr=M,/10+5 x2 = 1 (unit)
yz = y
x3 = 4 (unit)
y3 = MS/20 + 2.5
(49)
where y1 and y3 have been translated. The scale moduli for the three scales M,/lO, MJ20, and y are 1, 1, and 1, respectively. As long as Mp and M, are both larger than 20 dB, both scales are easily draw since (45) is valid (within 0.5%). Since the scale modulus for MS/20 is 1, the scale from 20 to 150 dB is drawn with 1 cm between major ,points (integer multiples of 10 dB). By the same method, we draw the Mr,/10 scale from 20 to 50 dB with 2 cm between major points (integer multiples of 10 dB). To check several points, notice that data points on our nomograph lie on horizontal lines (i.e. y1 = y2 = y3 in (49)) whenever the gains satisfy M,=2Mp+50,
M,>20dB
(50)
in which case M,>20dB
y=Mp/10+5=M,/20+2.5,
Also notice that whenever
M >gOdB s
the gains are equal, then from (44) -r=O
This condition must be satisfied forms another convenience check To obtain the scales for 20 dB for which the nomograph is being
for
M,=M,
(52)
by any nomograph for any (M,, MS) which point. and below for Mp and MS, we return to (44) constructed. The basic determinant of (44) is
0 log(lo”~lo-l) 1 Y 1 ~log(lo”“o-l) which reduces become
(51)
1 1 =o 1
(53)
to (47) for Mp, M, > 20 dB. Then the nomograph x1=0
yr=log(lO”~‘o-
x2= 1 (unit)
y2= y
x3=$(unit)
y3=ilog(10MJ10-1)+2.5
equations
1)+5 (54)
by analogy with (49). A Fortran program was written to obtain various (yr, y2, yJ) combinations from (54). These values allowed us to complete the Mp and M, scales. They were used for Mp, M, < 40 dB. This also verified that our assumptions leading to (45) were valid.
124
Journal of The Franklin
Institute
Nomographs and Filters
It is interesting to observe the behavior of yl and yz for MP, A4.<<10 dB. Using the Maclaurin series for ax as aX=l+xlna+
(x In a)’ 21 +***
where a = 10 and x = M/10, then for xc< 1, log (a” - 1) = log (x In a) = log x + log (In a)).
(56)
Since a = 10 and log (In 10) = 0.362, then from (54) Yl=log
&+ 1o
5.362 M,,ibf,<<
Yz=Y
10dB.
(57)
Therefore, y1 and y3 vary logarithmically for Mp and M, less than 1 dB. Since the scale modulus for log (&/lo) is 1, the Mp scale below 1 dB is drawn with 2 cm between major points (integer multiples of log (M&O)). Similarly, the A4, scale below 1 dB is drawn with 1 cm between major points (integer multiples of log (k&/10)). We see from (57) that y1 = y3 = 0 when Mp = i%f,= 10-4.362. We have completed our nomograph as far as the Mp, M,, and y scales are concerned. What remains is to draw a scale on the y support and to plot unique curves in the resulting grid. Recalling from (39) that y = log s:(Q)
(58)
s;(a) = @(“--m), R + a,
(59)
and observing that
for every rational filter having a gain of (23), then y=2(n-m)logO,
Q+w.
(60)
Thus, to produce asymptotically linear curves in the (y, a)-plane, we choose f-I to be a logarithmically-scaled axis. From experience, one cycle provides an adequate range for Q. This completes the construction of the nomograph in Fig. 7. Now we are ready to review the various classical filters and to plot their S,(Q) from (39).
IV. Classical Filter Polynomials Classical filters are filters whose magnitude response is related to a classical polynomial. They can be derived from a basic polynomial f of second degree where (16), (17) f(x) = (b-x)(x Vol. 302, No. 2, August 1976
- a)
61)
125
J. N. Hallberg and Claude S. Lindquist TABLE
III
Classical polynomials
(16)
General Jacobi polynomials f(X)=(b-x)(x-u)
P(x)=(b-x)‘(x-a)@
f(X)=(l-x)(x+1) =lPxz
simplification to the interval (-1, +1) b=+l I a=-I, Jacobi polynomials p(x)=(l-x)“(l+x)P simplification I a=@=h Ultraspherical polynomials p(i)=(l-x2)” simplification
f(X)=l-x’
by using the different weighting functions polynomials can be expressed in the form
S,(x) = ~
l
P(X)Kl
p listed
a>-1 p>-1 la\<00 lbl
cr>--1 p>-1 a=-1 b=+l A>-1
in Table
III.
All classical
(62)
~'"'[pW"Wl
where DC”) represents the nth derivative of the expression in brackets and K,, is which can be a constant which depends on n. The classical polynomials generated are listed in Table III. The most general polynomials are the hypergeometric polynomials F(a, b, c; x). With a, p > -1 and restricting the interfal (a, b) to (-1, l), the Jacobi polynomials result. If (Y= p, then these polynomials simplify into the ultraspherical or Gegenbauer polynomials. Since Jacobi polynomials satisfy p’,“.@‘(-w) = (_l)“p’,p’“‘(w)
(63)
and any polynomial P’, used in the gain expression (24) must be even, then (Y= /3. Thus, ultraspherical polynomials must be used. In order that F, have its monotonically for 1w I> 1, parameter zeros within 1wI < 1 and IF,,1 increase (Y> -1. (Y can therefore assume any value in the range -l
(64)
in Table IV. (Y= -0.5 produce the well-known Butterworth and filters, respectively. (Y= 0 produces the Legendre produces the modified n,. . . (any positive integer) introduced by Ku and Drubin in 1962 (18).Using - 1 results in the ultraspherical filters introduced by 1966 (19). They are credited for having pointed out were special cases of the ultraspherical filter. Journal ofTheFranklin
Institute
Nomographs and Filters TABLE IV. Classical filters based upon the ultraspherical polynomialwhich is based upon the moregeneral hypergeometricpolynomial
-4
Ultrospherlcol
Common
(cc)
Hypergeometrlc
nome
a3
Butterworth
IO
10th
M.A.L.
polynomials 3
F(U,b,CjX)
i
2 I.5
Jacob1 polynomials
n
PIUSO
) Lx) t 0.5
3rd
M.A.L.
2nd
M.A.L.
l-5
Ultraspherical
1st
M.A.L.
2nd
Cheybshev
lI Ultraspherical
0
‘-
Legendre
-0.25
--
-0.25
polynomials F,O
-05
--
1st Cheybyshev
-0.75
--
-0.75
-0-9
--
-0.9
--
-0.95
--
-0.9999
-0.95 - 0.9999
V. Ultraspherical
G(w) are formed
functions =
Ultraspherical Ultraspherica
I
Ultraspherical
(58). To find c(w),
given
by (24) where
F3w)12.
(65)
wxw)12
we use the recursion
(2a+n)F~(~)=x(2a+2n-l)F~_,(x)-(n-l)~-~(~) and
- Rwple)
by plotting Y = log
from
(Equal
Ultraspherical
Filters
Ultraspherical filters have magnitude
The nomographs
Ultraspherxal
(66) formula
n22
(67)
(19) F;;(x)= 1 C(x)
= x.
(68)
Since FE(l) = 1, (65) satisfies the frequency normalization condition of (29). A Fortran computer program was written to generate the ultraspherical polynomials (to orders 20) using (67) and (68), and y of (66). Nomographs were generated for the (Y values listed in Table IV and are shown in the Appendix in Figs. Al-A14. They are labelled with their common name and the value of alpha (CY)following it in parenthesis.
Vol.302,No.2,August 1976
127
J. IV. Hallberg and Claude S. Lindquist
Butterworth filters are the class of all-pole maximally-flat magnitude filters. They have no passband ripple and a magnitude function of
IHWI” =
1
+
(MFM)
e:w2”
where s;(w)= Since (S,(l)j=
W2”.
(70)
1, then from (58) y = log wzn = 2n log w.
(71)
These are straight lines of slope 2n so the Butter-worth filter nomograph is easily plotted using a straight-edge in Fig. Al. Using (Y= 10’ in (66) yielded the same results which were used to verify the ultraspherical filter program. Filter orders through 25 are shown. The modified associated Legendre (MAL) filters result for positive integer values of (Y.They use modified associated Legendre polynomials Pi”” which are obtained by differentiating an (n + m)-degree Legendre polynomial P,,+,,,m times where P’,“‘(x) = $
P,+,(x).
(72)
Excluding elliptic filters, they were the first 3-parameter filters where (n, E, a) could be selected. The lOth, 3rd and 2nd M.A.L. filters have the nomographs shown in Figs. A2, A3 and A4, respectively. Ultraspherical filters result for noninteger values of (Y> -1. The 1.5 ultraspherical filter has the nomograph of Fig. A5. The 1st M.A.L. filter has the nomograph of Fig. A6. 01 = 0.5 results in a Chebyshev filter of the second kind. This is not the common equal-ripple or Chebyshev filter (of the first kind) where (Y= -0.5. They have little in common except (Y’Sof opposite sign. They use Chebyshev polynomials of the second kind V,,(w). The nomograph is shown in Fig. A7. The Legendre jilters have (Y= 0. They use the Legendre polynomials P,,(w). Unfortunately, this filter is widely confused with the Papoulis or monotonic-L filter to be discussed later (20). The nomograph is shown in Fig. A8. The -0.25 ultraspherical filter has the nomograph shown in Fig. A9. This filter was discussed by Petrela and Budak (21). The Chebysheu jilter (of the first kind) or equal-ripple filter has (Y= -0.5. It uses Chebyshev polynomials of the first kind T,,(w). The nomograph is shown in Fig. AlO. The -0.75, -0.90, -0.95 and -0.9999 ultraspherical filters have nomographs shown in Figs. All-A14. These are not very practical because of the large ripple as shall now be discussed. The ultraspherical filters of this section (a = 00+ -1) have been ordered in terms of increasing bandedge selectivity, i.e., -d IHJ/dw I,,,=i. Several magnitude responses are shown in Fig. 8 where n = 10 and E = 1 (21). The filters 128
Journal
of The Franklin
Institute
Nomographs and Filters
Frequncy,
rod/see.
w
FIG. 8. Magnitude response of several 10th order ultraspherical
filters having E = 1 (21).
have 3 dB frequencies of unity since E = 1. When a B-0.5, the inband ripple increases as the passband edge is approached. For -1
Magnitude Filters
There are other classical filters which are not derived directly from ultraspherical filters and which must be considered separately. These include the Papoulis, Halpern, Bessel, Gaussian, and synchronously-tuned filters which we shall now discuss. The Papoulis filter, sometimes called the monotonic-l or L-filter, was introduced by Papoulis in 1957 (20). As mentioned before, it is often mistakenly called a Legendre filter. It combines the desirable features of the Chebyshev and Butterworth filters by having (1) monotonic passband response, and (2) the maximum bandedge selectivity possible. The polynomials L,(w2) of the Papoulis filter are related to and derived from the Legendre polynomials (22)-(24). Using L1-L8 from (22), its nomograph was plotted in Fig. A15. It is interesting to observe that the Papoulis filter nomograph is almost identical to that of the 1st M.A.L. Thus, their stopband responses are very similar (of course, their passband responses are quite different). Halpern introduced another monotonic response filter in 1969 (25). It is related to the Papoulis filter in the following way: Papoulis maximized the rate of change of IS,,(w)] at w = 1, (djS,(w)]/dwl,=l) under the constraint of response. Halpern maximized the asymptotic growth of monotonic IS,,(w)l ((S,(w)l/w”) under the constraint of monotonic response. The polynomials H,,(w’) of the Halpern filter are related to and derived from the Jacobi Vol.302,No.2,August 1976
129
J. N. Hallberg and Claude S. Lindquist
polynomials. Using HI-H7 in (25), the nomograph was plotted in Fig. A16. It can be shown that the improvement in stopband rejection of the Halpern filter over that of the Papoulis filter is about OSJn. Since this is only about 1.7 or 4.7 dB for n = 10, the two nomographs are about the same except for large n. VII. Linear Phase/Constant
Delay Filters
The filters which have been considered so far have approximated magnitude response of the ideal lowpass filter having gain
the
H(jw) = eviTow, 1w I< 1
(73)
= 0,
Iwl’l
which equals unity for 1w ( < 1 and zero elsewhere. Now we shall consider filters which approximate the phase and delay response of the ideal filter where
Iwl
arg H = - WT~,
= arbitrary,
Iw I > 1
(74)
d arg H r=
Iwl
-dw=rO~
= arbitrary,
Iw I > 1.
These are called linear phase or constant delay filters. The Bessel or Thomson filter is the class of all-pole maximally-flat delay (MFD) or linear phase filter (26), (27). They approximate delay in the same way as Butterworth filters approximate magnitude. They were introduced by Thomson in 1949. Their transfer functions equal
H,(s) = where the polynomials by (28)
_ Qn(O)
a0 ao+aIs+.
. .+a,s”
(75)
Q,(s)
Q,(s) are obtained from the Bessel polynomials
Q,(s) = s”J%(l/s).
B,(s) (76)
It is important to note that this is different than say, for example, the Chebyshev or Legendre filters which use their respective polynomials directly. Thomson gives the squared modulus polynomial as
Q(lljw)Q(-lljw)
= f B,~(w*)~
(77)
k=O
where (n + k)! (2k)!
(78)
Bnk=(n-k)![k!2k]2.
Wood gives the same result and the useful recursion formula (29) & 130
= (2n - 1)(2k - l)Bn-r,k-i
+ B,,-_z,k
(79) Journal
of The Franklin
Institute
Nomographs
and Filters
where
Bno= 1 B
nl
(80)
=n(n+l> 2
.
A Fortran computer program was written to generate the Bessel filter polynomials to order 20. Using these polynomials, the squared-magnitude function can be written as
bo
IfL(j
= bo+blW2+.. .+bnW2n 1 1+;w2+.. 0
Therefore,
from
(81) .++w2" 0
(24), then
s’,(w)=+ w*+.. .+; W2n. 0 1
(82)
Frequency normalization by w3 dB is required to force (S,(l)/ = 1. w3 dn was found using iterative techniques. The nomographs for the Bessel filter is shown in Fig. A17. For n large and w small, the filter is Gaussian (28) as we shall verify. The Gaussian filter was introduced by Dishal in 1959 (30). The magnitude response of the ideal Gaussian filter equals IH(
=
l/IM(jw)l
where IH( =$. Expanding /HI’= truncating at the nth term gives
=
e-“~347w2= e-aw2’2
exp (-aw2)
(83)
in a Maclaurin
IM(jw)l’ = eaw2g l+2(aw)2+g(aw)4+.
. .+$
(aWyn
series
and
(84)
which is the squared-magnitude response of an nth order Gaussian filter. A Fortran program was written to generate these polynomials to order 20. Since the magnitude response has the form of (81), then S”, is found by using (82). Frequency normalization by wj dB is required to force (S,( l)( = 1. wsdB was solved for using iterative techniques. The nomograph for the Gaussian filter is shown in Fig. A18. As n + m, the response becomes truly Gaussian. For large w, then
S2(w) = e2(o.347wz),1w 1>>1 Vol. 302, No. 2, August 1976
(85) 131
J. N. Hallberg and Claude S. Lindquist
and y=10gs2=0.3w2,
}w(>>l.
(86)
This is the limiting curve as n + 00 in the nomograph. Thus, as with the Bessel filter, there is a maximum rejection which can be obtained independent of order. Note that the Bessel nomograph approaches the Gaussian nomograph for large n and w = 1 (this is the “peculiar” cross-over effect). Synchronously-tuned filters have the simplest transfer function of all filter types. They consist of n identical cascaded filter stages all having the same simple pole po. They have a squared-magnitude function of
IHntiw)l*= ,l+~~~~. To maintain unity 3 dB bandwidth,
the pole p. must be located as
J(n)
Po=(2”“-l)-f+__
J(ln 2) a Using the binomial theorem
(87)
W
to expand IMn(iw)12 as
where the coefficients of (w/pa)* are the well-known binomial coefficients. since (90) a Fortran program was written to calculate y for orders to 20. The resulting nomograph is shown in Fig. A19. Noting that for large n, the magnitude response can be expressed as IHO.w)l=!?_[l+$]-“‘*
(91)
where 0 = J(ln 2) w, then making the change of variable v= n/Cl, we obtain
(92) which is Gaussian from (83). Thus, the synchronously-tuned filter also has the same limiting y as the Gaussian filter. It may be questioned why we treat the 132
Journal of The Franklin Institute
Nomographs and Filters synchronously-tuned filter as a constant delay filter. The reason is that its selectivity is low, its delay response is fairly flat, and its step response is monotonic (2) which is similar to the other constant delay types. VIII. Usefulness of Filter Nomographs We have reviewed a variety of classical filters and presented a collection of nineteen nomographs which describe their magnitude responses. We can easily group these filters in two main categories as: (1) High selectivity filters (2) Low selectivity filters based on the shape of the y = log S’, curves. Using the linear curves given by (60) as a reference (e.g. the Butterworth filter curves), curves convex above the reference are highly selective; those convex below the reference (or concave) have low selectivity. The ultraspherical filters are highly selective (but large ripple) for (r = -1. They become decreasingly selective for increasing cz + 00. The Papoulis and Halpern filters had selectivities close to those for 1.0 ultraspherical filters. The constant delay filters; Bessel, Gaussian, and synchronously-tuned filters, were very selective (increasingly so in this order). This gives a much more general interpretation of selectivity than has been available in the past. It also allows filters to be easily compared. The nomographs make the determination of filter order trivial. For example, suppose that the filter orders must be determined to meet the response of Fig. 9. Expressing the parameters in the form of Fig. l(a) as M,=ldB M,=40dB
at
R=l.S
(93)
we enter this data on the nomographs in the Appendix as shown in Fig. l(b) (Y = 4.6 and fi = 1.5). Then the minimum filter orders are listed in Table V. We see that none of the constant delay filters can be used. We can use a 14th order Butterworth,,8th order Papoulis, 8th order Legendre, or 7th order Chebyshev
FIG.
9. Example of required response of a lowpass filter.
Vol. 302, No. 2. August 1976
133
J. N. Hallberg and Claude S. Lindquist TABLE V Minimum order filters for filter response of Fig. 9
Filter type Chebyshev (-0.5) -0.25 ultraspherical Legendre (0) Chebyshev (0.5) Halpern Papoulis 1st M.A.L. (1) 1.5 ultraspherical 2nd M.A.L. (2) 3rd M.A.L. (3) 10th M.A.L. (10) Butterworth (m) Bessel Gaussian Synchronously-tuned
(equal-ripple) filter. order via equations nomographs.
Minimum order n 7 7 8 8 8 8 8 9 9 10 11 14 Impossible Impossible Impossible
Any engineer who has wrestled with determining filter will immediately recognize the tremendous value of
IX. Comparison with Earlier Work We mentioned earlier that (1)and (2) contain certain inaccuracies. These can produce errors when determining order which result in filter designs which do not exhibit the required stopband rejection. This is of course puzzling when the inaccuracies are not recognized. Therefore, it is important to identify the problems. Kawakami (1)used analogous equations for y [see Eqs. (39) and (44)]. His plots for y = log S’,(n) = r(n) appear to be correct. However, the y-axis has been shifted slightly (to the right for Butterworth and the left for Chebyshev). This may be verified by checking y = 0 for several Mp = M, combinations [see Eq. (52)]. These errors appear to be compounded in (2). We see that it is vitally important to insure the exact geometric relation required by (20) for scale spacing, (53) for marking of the scales, and (39) for plotting y vs. 0. These observations prompted us to design our nomograph of Fig. 7 with maximum symmetry. We were careful to choose scale factors m, and m, of unity so (x1, x2, xg)= (0,1, $) in (20). References (1) and (2) use about (0 1,0.56). With our choice of scale factors, (u, U, w) = (M,/lO, y, M,/20) (assuming Mp, M, >>10 dB) all had unity scale factors which insured equal spacing between major points of i’$/lO (10 dB steps), MJ20 (20 dB steps), and y (unit steps). By translating (u, U, w) so that (50 dB, 10, 150 dB) all lie on the
134
Ioumal of The Franklrn
Institute
Nomographs same
choice
and Filters
horizontal line, the scales were easily marked off and completed. of scale factors in (1)and (2) did not yield this convenience.
The
X. Conclusions Filter nomographs provide a graphical means for determining filter order and also tradeoffs in maximum passband attenuation MP, minimum stopband rejection MS, normalized stopband frequency &I, and order n. They also allow the stopband responses of various filters to be compared. Thus, they provide maximum information with a minimum of time invested. The basic nomograph axes are easy to construct once basic nomograph theory is understood. The same axes can be used for any filter. Only the y = -y(a) curves must be modified to match the standard response functions being used. The equations which describe these curves are both complicated and long so they require computer evaluation. However, once the nomograph has been obtained, it can be used for all future design. This paper has presented the nomographs of 19 classical filters of which 17 nomographs are new. The other two nomographs replace the earlier nomographs which contain inaccuracies. References (1) M. Kawakami, “Nomographs for Butterworth and Chebyshev filters,” IEEE Trans. Circuit Theory, Vol. CT-lo, pp. 288-289, June, 1963. (2) A. I. Zverev, “Handbook of Filter Synthesis,” Wiley, New York, 1967. (3) E. Christian and E. Eisenmann, “Filter Design Tables and Graphs,” Wiley, New York, 1966. (4) J. N. Hallberg, “Filters and nomographs,” M.S.E.E. Directed Research, California State Univ., Long Beach, Ca., Jan., 1974. (5) A. Budak, “Passive and Active Network Analysis and Synthesis,” Houghton Mifflin, Boston, 1974. (6) M. S. d’ocagne, “Traite de Nomographie,” 2nd ed., Paris, 1921. (7) L. Johnson, “Nomography and Empirical Equations,” Wiley, New York, pp. 19-33, 1952. (8) H. J. Allcock, “The Nomogram,” 5th ed., Pitman, London, 1963. (9) L. I. Epstein, “Nomography,” Interscience, New York, 1958. (10) J. W. Richards, “Introduction to Graphs and Nomograms,” Heywood Books, London, pp. 98-110, 1966. (11)A. S. Levens, “Nomography,” Wiley, New York, 1959. (12) R. D. Douglass, “Elements of Nomography,” McGraw Hill, New York, 1947. (13) J. H. Fasal, “Nomography,” Ungar, New York, 1968. (14) E. Otto, “Nomography,” MacMillan, New York, 1963. (15) D. Davis, “Nomography and Empirical Equations,” Reinhold, New York, 1962. (16) J. Vlach, “Computerized Approximation and Synthesis of Linear Networks,” Chap. 6, Wiley, New York, 1969. (17) M. Abramowitz and I. A. Stegan (Eds.), “Handbook of Mathematical Functions,” Natl. Bur. of Standards, Vol. 5.5, Sect. 22, 1972. (18) Y. H. Ku and M. Drubin, “Network synthesis using Legendre and Hermite polynomials,” J. Franklin Inst., Vol. 273, pp. 138-157, Feb., 1962. filters using ultraspherical (19) D. E. Johnson and J. R. Johnson, “Low-pass mials,” IEEE Trans. Circuit Theory, Vol. CT-13, pp. 364-369, Dec.,
Vol. 302, No. 2, August 1976
polyno1966.
J. N. Hallberg
and Claude S. Lindquist
(20) A. Papouiis, “On the approximation problem in filter design,” IRE Natl. Cow. Rec.. part 2, pp. 175-185. 1957. (21) D. M. Petrela and A. Budak, “Pole locations for ultraspherical filters,” IEEE Tram Circuit Theory, Vol. CT-17, pp. 668-670, Nov., 1970. (22) A. Papoulis, “Optimum filters with monotonic response,” Proc. IRE, Vol. 46, pp. 606-609, March, 1958. (23) A. Papoulis, “On monotonic response filters,” Proc. IRE, Vol. 47, pp. 331-333, Feb., 1959. (24) M. Fukada, “Optimum filters of even orders with monotonic response,” IRE Trans. Circuit Theory, Vol. Cl’-6, pp. 277-281, Sept., 1959. (25) P. H. Halpern, “Optimum monotonic low-pass filters,” IEEE Trans. Circuit Theory, Vol. (X-16, pp. 240-241, May, 1969. (26) W. E. Thomson, “Maximally-flat delay networks,” IRE Trans. Circuit Theory, Vol. CT-16, p. 235, June, 1959. (27) W. E. Thomson, “Delay networks having maximally flat frequency characteristics,” Proc. IEE, pt. 3, Vol. 96, pp. 487-490, Nov., 1949. (28) L. Starch, “Synthesis of constant time delay ladder networks using Bessel polynomials.” Proc. IRE, Vol. 42, pp. 1666-1675, NOV., 1954. (29) I. E. Wood, “Note on maximally flat delay networks,” IRE Trans. Circuit Theory, Vol. CT-5, pp. 363-364, Dec., 1958. (30) M. Dishal, “Gaussian response filter design,” Electrical Commun. Vol. 36, No. 1, pp. 3-26, 1959.
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136
(m) filters. Journal of The Franklin Institute
Nomographs and Filters
Vol. 302, No. 2, August 1976
137
J. N. Hallberg and Claude S. Lindquist
138
Journal of The Franklin
Institute
FIG. A6.
Nomograph for first modified Legendre (1) filters.
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140
Journal of The Franklin
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Nomographs and Filters
J. IV. Hallberg and Claude S. Lindquist
142
Journal of The Franklin Institute
Nomographs and Filters
Vol. 302, No. 2, August 1976
143
J. N. Hallberg and Claude S. Lindquist
I I I 000000000 LOZ’OEZQrnW
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