- Email: [email protected]
A new type of wave digital filter
A New Type of Wave Digital Filter by M. N. s. SWAMY
and K. s. THYAGARAJAN
Department of Electrical Engineering Concordia University *, Montreal, Canada
ABSTRACT
LC-ladder equivalent
:
This paper
proposes
networks by replacing digital two-port.
from doubly terminated amlEelement of the ladder by its
a new wave digital filter derived each series
or shunt
It is shown that such two-ports
may be cascaded without the use
of adapters defined by Fettweis
(1).A number of realizations of the wave dig&al two-ports, which are canonic with respect to both multipliers and delays, have been obtained. Also a realization
which
is canon&
with respect
considered
using this realization.
coeficient
change8 due to finite
to multipliers
The sensitivity
word length is compared
digital filter and also the one proposed
is given
and an example
with the conventional
by Renner and Gupta. It is
to be a more desirable form of implementation and cornparable to that of Renner and Gupta (2).
Jilter appears form
only
of this jilter with respect to the multiplier
found that the
than the conventional
cascaded proposed
cascade
I. Introduction
The design of digital filters which have low sensitivity to multiplier coefficient variations is currently receiving attention. Ideally it would be desirable to realize a digital filter whose transfer characteristics, namely the magnitude and phase responses, remain invariant to changes in the multiplier coefficients due to their finite word lengths. Under such situations, the length of the coefficient register can be kept to a minimum thus resulting in reduction in the cost of hardware units. To this end, Fettweis (1) has proposed a class of filters known as wave digital filters which exhibit much less variation in the transfer characteristics due to multiplier coefficient changes than the conventional form of digital filters do. Using Fettweis’ technique, Renner and Gupta (2) have given procedures for designing digital filters patterned after maximum available power (MAP) networks. These filters are, in general, non-canonio with respect to the multipliers. However, due to the low sensitivity properties, the lengths of the coefficient registers for the wave digital filters need be smaller than those of the conventional digital filters. This decrease in the coefficient register lengths in wave digital filters may not only result in a decrease in computational time between samples but also in a reduction in the overall cost of the hardware units. The design of wave digital filters proposed by Fettweis and others (l-5) is based on imitating certain doubly terminated LC-ladder structures. Instead of the voltages and currents, incident and reflected voltage or current waves are * Incorporating Sir George Williams University and Loyola of Montreal.
41
M.
N.
S.
Swamy and K. S. Tlzyagarajan
used as the input and output signals in wave digital filters. Using a transmission line transformation, each two-terminal element in the analog filter is transformed to a corresponding two-terminal wave digital element with the incident and reflected waves as digital input and output. However, direct interconnection of these wave digital elements is not possible in general due to incompatibility of the waves between the elements. So digital “adapters” are used between the elements so that the waves are made compatible between the different wave digital elements. The use of these adapters in wave digital filters, however, seems to make the design of such filters more complicated. This paper considers a different design approach to wave digital filters starting with doubly terminated LC-ladder networks, wherein the different circuit elements are represented as wave digital two-ports. It will be shown in a later section that the wave two-port characterization of the analog elements makes it possible to interconnect the different ports directly without the use of adapters. This makes the design procedure very simple. A number of different canonic realizations for the wave digital two-port is given. Also, an example is taken to exemplify the sensitivity properties of one of the realizations and those of the filters realized in the usual cascaded form or by Renner and Gupta’s method (2).
ZZ. Characterization
of the
Wave
Digital
Two-port
Consider Fig. 1, where the black box 8 may contain an interconnection of linear elements. If we now define the wave variables (&, gi), i = 1,2, by
(1) where R, and R, are arbitra,ry port resistances at ports 1 and 2, then we may express (& &) in terms of (&, a,) as (6)
where A+R,C+B+R,D p=---F-
2R,’
x = A+R,C 2
B+R,D -
2R,
’ >
fi=
R=
42
(3)
A-R&B-RID 2
2R,
A-R,C
B-RID
2
-
2R,
’
I
Jo:mal
of The
Franklin
Institute
A New Type of Wave Digital Filter
and chain matrix of m. II--c -..
-
1
(4) 12
LINEAR NETWORK
,
Rz
1
I
FIG. 1.
It may be observed that the determinant of [p] is
~F~=$(AD-BC). 2
In general, the elements of [p] are functions bilinear transformation (7) a=-
of S. If we now use the
z-1 z+l
and identify a”, and a”, with the input, and gl and g2 with the output variables of a digital two-port, then we may recognize (2) as the chain matrix description (8) of a digital two-port. We shall write (2) with (6) substituted as
Kj=[‘:11[I =[Fl[:j,
where
[F]=[P]
withs=-.
z-1
z+l
(74
Pb)
If N is reciprocal, it is seen from (5) and (7b) that IFI =$. 2
(8)
Thus, given any two-port network m with port resistances R, and R,, ale may describe a corresponding wave digital two-port N, whose chain matrix is given by (7). It is clear that if fi is a cascade of two-ports, say, fll, R2, . . . , iV,, then the corresponding digital two-port is the cascade of the digital two-ports N&N,, . . . . N, where Ni is the digital two-port corresponding to fli ; of course, it is assumed here that the output port resistance of fld is the same as the input port resistance Of lo,+l. ZZZ. Realization
of the Digital
Two-port
We shall now consider the realization of a digital two-port starting with a doubly terminated lossless network fl (Fig. 2), since it is known to have very
Vol.
300, No.
1, July
1975
43
M. N. S. Swamy and K. S. Thyagarajan
T (5) = V, /V, FIG. low
2.
sensitivities with respect to the different elements. It is seen from Fig. 2
that v, = v,-JIBR,,
(9)
v, = -12RRL.
(10)
Hence, from (1) and (lo), we have d, = J,$
(11)
and g, =
2Ji
(12)
l+d’
where +=-.
h--R, R,+R,
(13)
= (l+e)a,,
(14)
Also, from (1) and (9), we have a”,+&, where v, = CSi,,
(15) (16)
Hence, 6, 2% Ej=(l+fj)~=iq~*
2
v,
(17)
Also from (2) and (14), (p+Xl#&
= (l+e)G&--e(~+k~)9,
or
If we denote
v,
H(z) = v,
z-1
with s = z+l
(19)
then b 2 -?- -H(z), a, 1+$,
44
(204
Journal of The Franklin Institute
A New Type of Wave Digital Filter where
a2 =
WC)
+b,,
a, = (l+O)a,-Bb,. The corresponding realization two-port corresponding to fl.
Pod)
is shown in Fig. 3, where N is the digital
I=- RL - Re RL + Rz
it&= ?aI ‘+o
H (4
t
H (z) = -
V?.
With
I = g
VS
FIG. 3.
Thus, given the realization for T(s) = (I?&) as a lossless two-port can obtain the corresponding realization for H(z) = [V,/V,] with S = [(z-
l)/(z+
fl, we
l)]
by using Fig. 3, where the digital two-port N is obtained from the corresponding LC-network 8. Until now, we have not made any assumptions about 19. However, in order to realize N from &’ in a simple fashion, we shall assume fl to be a lossless ladder. Then the series elements can only be an L, C or an LC-parallel circuit, while the shunt elements are L, C or a series LC-circuit. Thus, it is necessary to derive the digital two-ports corresponding to only these elements, which we shall do in succeeding sections. Then, we may cascade them to form N and use Fig. 3 to realize H(z). IV.
Design for
of Digital
all the Series
Two-ports
Corresponding
and Shunt
Elements
to the Case
R, = R,
in m
Let a series element in fl be denoted by ZJs), while a shunt element by Y,(s). Let 2, and Y, represent the functions with s = (z- l)/(z+ 1). Then, for a series element Z,, the digital two-port description is
(21)
Vol.
300,
No.
1, July 1975
45
M. N. S. Swamy and K. S. Thyagarajan with 111 =
VI =
2, + (R, + R,) 2R, ’
(22)
2, + (4 - R,) 2R, ’
& = (I-PI),
‘Q = (I-“,),
while for a shunt element Yb (23) where
x
=
V2 =
(24)
Y2+(Gl-Gz) 2G,
2
’
(1 -_cL2), K2 = (1 -A,).
In the simple case when R, = R, = R (say), (21) and (23) may be rearranged as
b, =
I
a,+~(al---a,),
b, = al-:(a,--a,) for the series element, and
(25)
I
b, = a,-~(a,+a,),
(26) b, = a,-~(aI+a,)
I
for the shunt element. For illustration, the corresponding realizations are shown for series L and shunt C in Fig. 4(a) and (b). Realizations for series C and shunt L may be obtained by changing z-l to -z-l and L to (l/C) in Fig. 4(a) and (b) respectively. Similarly, realizations may also be obtained for a parallel LC in the series arm and a series LC in the shunt arm. It should be pointed out that there are no delay-free loops in any of these realizations. However, when these are cascaded, delay-free loops appear, a situation which is similar to the one encountered in wave digital filters discussed in Refs. (1) and (2). In order to overcome this difficulty, we introduce an extra delay element either in the forward path or in the return path, where two sections are cascaded. It should be pointed out that one can always choose either R = R, or R = R, in Fig. 2, so that, in Fig. 3 either $ = 0 or 0 = 0.
46
Journal of The Franklin
Institute
A New Type
of Wave
Digital
Filter
A design example which is considered in Ref. (2) will now be taken. Let it be required to design a third-order low pass Butterworth filter with a cut-off frequency of 100 Hz and sampling frequency of 10 kHz. Since
R
@
!/L +_
:
c
+
_t
2-l
I
_+
@ R
I
oa2 <=
(2R-L)/(2R+L) (a)
b,
0
02
a=(ZG-C)/(ZG+C)
(b) FIG.
bilinear transformation prewarped using
is being
4.
used,
the
critical
w, = tan(+,T),
frequency
must
be (27)
where wA and wD correspond to the critical frequencies of the analog and digital filters, respectively. The analog filter is shown in Fig. 5 with the
“S:l”aTG--Jj
T(4= v,/ V, r WA= 0.031426266 FIG. 5. corresponding digital filter in Fig. 6. In order to examine the effect of quantization of the multiplier coefficients, plots of the magnitude response of the wave digital filter of the above example are shown in Fig. 7 for different levels of fixed point quantization. To make a valid comparison the same digital transfer function is implemented in the conventional cascade form
Vol. 300, No. 1, July 1975
47,
M. N. 8. Swamy and K. S. Tiiyagarajan as shown in Fig. 8, and its corresponding plots are shown in Fig. 9. From the two plots, we observe that the shift in the magnitude response due to coefficient changes is less for the proposed wave digital filter than that for the conventional cascade filter. It is found that for this example, realization is possible even with word lengths of 5 bits for multipliers for the proposed
da = -0.88172593,
4
= -0.939062506,
FIG.
ds = 0;
6.
0
-1 -1.5 * .s
-2
f f
-2.5
p -3 t
-3.5 -4 -4.5 -5 0
10
20
30
40
50
60
70
80
90
100
110
-_,H.Z
FIG.
D(l+z
4
-=
01
( I + AZ-’
-1 )3
) ( I + Bz-’
FIG.
48
7.
+ CZ’~
)
8.
Journalof The FranklinInstitute
A New Type of Wave Digital Filter wave digital filter while a minimum of 15 bits are required for the conventional cascade filter and 10 bits for the filter proposed by Renner and Gupta (2). The different parameters of interest are listed in Table I. From the entries in 0.5 0 -0.5
-1 -1.5 B .5 -2 0 3 -2.5 .z P E f
-3 -3.5 -4 -4.5 -5
I
I
0
10
20
30
40
50
FIG.
60
70
80
90
1M) *HZ
110
9.
TABLE I Cascaded $.?ter L 3 dB frequency (Hz) o/o error Gain at 0 Hz (dB)
15
16
1’7
18
89.5 - 10.5 0.425
100 0 0.391
99.8 - 0.2 0.409
99.9 -0.1 0.400
19 100 0 -0.16
00 100 0 0
Renner and Gupta’s jilter y 3 dB frequency (Hz) o/o error Gain at 0 Hz (dB)
15
16
17
18
19
co
99.8 -0.2 - 0.0668
100.2 0.2 - 0.0053
99.9 -0.1 - 0.0293
100 0 0.005
100 0 0.005
100 0 0
Proposed filter y
5
3 dB frequency (Hz) o/o error Gein at 0 Hz (dB) % error = f+.
m
106.8 6.8 5.7
15 99.9 -0.1 -0.052
16 100 0 0.0151
17 100 0 -0.0174
18 100 0 -0.00118
19 100 0 -0.000917
co 100 0 0
100.
Vol. 300, No. 1, July 1975
49
M. N. S. Swamy and K. S. Thyagarajan the table, we observe that the proposed wave digital filter is more desirable than the conventional cascade implementation. It may also be noted that the proposed wave digital filter is comparable to that of Renner and Gupta.
V.
Canonic
realization
of Wave
Digital
Two-ports
Until now, it had been assumed that the port resistances R, and R, were equal for each of the individual two-ports in m. Under this condition, even though the corresponding digital two-ports had no delay-free loops, cascade of such two-ports had delay-free loops. In order to remove these loops, we had to introduce extra delays. We shall now show that by relaxing the condition R, = R,, we can obtain digital filters which are canonic with respect to both delays and multipliers. The digital two-port descriptions corresponding to series and shunt elements are respectively given by (21) and (23) and may be rewritten as 6, = a2 + (4~~) (a, - a2), b2 = a2 + (I/A
(a, - a2)
1
for the series element, and 6, = a2 s + (44
(a, + u2a,),
b2 = - a2 + (I/p2) (a1 + o2 a,),
(Jo= 4/R,
I
(29)
for the shunt element. These may be realized as shown in Fig. 10(a) and (b). It is seen from these figures that there are delay-free paths from a2 to b, in
2
(4
@) FIG. 10.
the realization of both the series and shunt elements. Thus, if (vJpl) and (v2/p2) are made to have no delay-free paths, then there will be no delay-free paths from a, to b1 ; as a consequence, there will be no delay-free loops when two sections are cascaded. In order to ensure this, we should make sure that the numerators of (vl/pl) and (v2/p2) should be of at least one degree lower in z than their corresponding denominators. We shall first show how we can achieve this for a series L. For this element, we may write
z(R,-R,+L)+(R,-RI-L) ;I = z(R,+R,+L)+(R,+R,-L)’
50
(30)
Journal of The Franklin
Institute
A New Type
of Wave
Digital
Filter
To ensure (vr/pr) to have no delay-free paths, we make (R,-R,+L)
= 0
In this case, the digital two-port
or
R,
description
= (R,+L).
(31)
becomes
(32) b, = a,+o
where
R,
R,
(33)
a=iTl=(Rz+
The realization for (32) is shown in Table II. Table II also shows realizations when the series element is a C or an LC-parallel circuit. TABLE
First
set of realizations
corresponding
II
to the series from a, to b,
SAME AS NETWORK
N, WITH
Z
element
REPLACED
2,
BY (-Z-
w&h no delay-free
puth
)
We shall now consider the shunt element; the chain matrix [FP] of the corresponding digital two-port ND is given by (23). Let N,’ represent a digital two-port, whose transfer matrix is the transpose of that of N; such a two-port can be obtained from N by simply reversing the arrows in N. Then the chain matrix of N,’ is given by
[F,l, =
$[“”“1, v2
Vol.300,No. 1,July1975
(344
K2
51
M. N. S. Swamy and K. S. Thyagarajan where A=]F,]=$?
(34b)
2
If we now identify Ya with Z,, G, with R, and G, with R,, we see that (34) reduces to --A, [-r(‘,]k = P1 . (35) [ -Q Kl I The realization for the above is the same as that of 2, with b, changed to -b, and a2 to -a2. Hence, the realization for the shunt element Yb may be obtained by starting with the realization of the series element for which 2, = Y6, transposing it, changing b, to - b,, a2 to - a2 and replacing R, by G, and R, by G,. Thus, for a shunt capacitor C, the realization is as shown in Table III, where u = (G2/G1), G, = G,+ C. The realizations for an L and a TABLE III
First set of realizations ELEMENT
corresponding to the shunt element Yz with no delay-free path from a, to b, CORRESPONDING
WAVE
NETWORK SAME AS NETWORK
N4 WITH
DIGITAL
RELATIONS
2 -PORT
??
N ‘, Z -’
NETWORK
REPLACED
N5
BY ( -2-l
d,Gr
)
GI G, = Gz + I a-=
GZ r !
d=
I - LC ItC
G, = Gz + ,&
0
I NETWORK
N6
series LC circuit in the shunt arm are shown in Table III. It should be observed that there is no delay-free path from a, to b, for any of these realizations. Thus, any of the networks N,-N may be cascaded without delay-free loops. It is also seen that if networks N,-N, are used, we may always choose the port resistance R, of the last element in the lossless network to be load resistance RL. We shall refer to the corresponding digital realization as realization I(a). We may obtain an alternate realization for the series element (and hence for the shunt element), and thus obtain an alternate realization for the
52
Journal of The Franklin Institute
A New Type of Wave Digital Filter digital filter. Consider the series element for which the chain matrix [F] of the corresponding digital two-port N, is given by (21). Let [Fr] be the chain matrix of the two-port obtained by reversing the ports of N,. Then, (36) From (21) and (36), it is seen that [F,] = [F] if we interchange the roles of R, and R,. Thus, a second realization for 2, can be obtained by simply reversing the ports of the realizations given in Table II and interchanging R, and R, in those realizations. These are shown in Table IV. Similarly, a TABLE IV
Second
set of realizations
corresponding to the series from a2 to b, CORRESPONDING
ELEMENT
WAVE DIGITAL
element
2 - PORT
2,
with no delay-free
path
RELATIONS
g=- RI b ;gi-Jy]~g
A: $ a
R2 = R, + L
0
6=-
RI R2
F? 0
R2 = R,
+-
0
Rz = R,+--I 1 LC
second set of realizations corresponding to YZ may be obtained using Table III and are shown in Table V. It is seen that when these networks are cascaded, there would be no delay-free loops. Further, if these are used in obtaining N for Fig. 3, we may always assume that the port resistance R, of the first element in the lossless network 8 is equal to R, in Fig. 2. We shall refer to the corresponding digital realization as realization II(a). For the sake of illustration, consider the network of Fig. 11. Then, using networks N,-N, we may obtain realizations I(a) and II(a); these are shown in Figs. 12 and 13. In Fig. 12, the different networks are N,:
network N,
with a = G,lG,,
N, :
network N1
with a = RJR,
NC:
network N4
with CJ= G,/G,
Nd:
network Ns
with u = RJR,,
Vol. 309, No. 1, July 1975
ar= (1 - L4 C,)/( 1 i- L, C,), (37) 01= (1 - L, C,)/( 1 + L,C,)
53
M. N. 5. Swamy
and K. S. Thyagarajan TABLE V
Second set of realizations corresponding to the shunt from a2 to b,
I
ELEMENT
CORRESPONDING
element Yz with no delay-free path
WAVE DIGITAL
2 - FORT
G2 =G,+C
.j= -G, G.2
G2 =G,++
G2
c = GI + I+L~,
I
I
FIG. 11. Nd
Rc
Nc
_^
Rb
Nb
R,
N,
G,
C4 = GL + I + LqC*
I$ = R, + L,
G,
= Gb + Cz
%=R=+j&
@=Rd-R, Rd + R,
$=
H, (L)
RL -0
_c
bz
cIz=o
a 1 = 2 H (z)
FIG. 12. bz 0 :
bz-
-
RA 9-
NA
RB
NB
RC
NC
RD
L
GC=GD+Ca
I
+ +,
Rb=RC+Ls +=$$&
RS
___a===%
R~ ‘Rs
ND
01
GA=
, ,
:=
C4 GB + I + L4 c4
Ha(z)
=++
H(z)
FIG. 13.
54
Journalof
The Franklin
Institute
A New Type of Wave Digital Filter and the function realized is b 2 = H,(z) = 2H(z), a,
(38)
where z-l
H(z) = 3
with s = -
Zfl
8
.
(39)
In Fig. 13, the different networks are N,:
network
hT, with a = R,/R,,
N,:
network N4
with (T= CD/G,,
NB:
network Nr
with (5 = RJR,,
N,:
network N6
with c = GBIGA,
and the corresponding
a! = (l-L,CJ(l
+L,C,), (40)
(II= (1 - L4C4)/( 1 + L,C,)
I
function realized is b -2 = H,(z) = -%
H(z)
(4la)
l+d
a1
with
4= VI. Alternate
Canonic
R,-RA
(41b)
R,+R,’
Realizations
We have shown in the previous section that we can always get two canonic realizations for the digital filter of Fig. 3, starting with the analog network of Fig. 2. In both these realizations, N is a cascade of subnetworks whose chain matrices are of the form (7) with (F 1= (RJR,). We have termed these realizations I(a) or II(a) depending on whether Tables II and III or IV and V have been used. We shall now show that we can always get three other realizations corresponding to each of I(a) and II(a). We call these realizations (b), (c) and (d), respectively. Realization (b) Let the realization (a) [I(a) or II(a)] be as in Fig. 3. Let the chain matrix of N be [F] as given by (7). Hence, we may express [;:I
(42a)
= q]
where R,P
[F,l=Iz;
[
_)I
-v
K * I
(42b)
If we now put the constraints
Vol. 300,
No.1,July
1975
a, = - Ob,,
(434
a2 = (I-+)a,++b,,
(43b)
55
M. N. S. Swamy and I<. S. Thyagarajan then
Using (13), (20a) and (2Ob), we may reduce the above to
The corresponding
b R 1 -1=1 RL r8 zH(z). as realization is shown in Fig. 14.
b, _=-. 0s
RI RL
is*
(44)
2 H(z)
FIG. 14.
Realization (c) Let NT be the digital two-port whose transfer matrix is the transpose of that of N. This may be obtained from N by simply reversing the direction of the arrows in N. Then the chain matrix of NT is
where A = IF]. If [F] is given by (7), then
[F,] =
2f [
h
I.
(45b)
K
Thus, if N is replaced by NT in Fig. 3, we see that b R 1+e AC-2 R,p+h$+~B+~hj as or
(46)
This will be referred to as realization
(c) for H(z).
Realization (d) Similarly, it may be shown that if N is replaced by NT in Fig. 14, we get b = R 2 A 2 R, G-i3 H(zL as which we will refer to as realization (d) for H(z).
56
(47)
Journal of The
Franklin
Institute
A New Type of Wave Digital Filter As an example, let us obtain realizations I(b), I(c) and I(d) from the realization I(a) given in Fig. 12. These are shown in Fig. 15(a), (b) and (c). Similarly, realizations II(b), II(c) and II(d) may be obtained from II(a) of Fig. 13. a,-
-
b,
__
Rd Nd R,
NC
Nb
lb
b
R~
Na
02
A
b,
-= 01
Rd -*RL
1. 1+e
R,
N,
H, (L)
= R
* &
* 2 H (2)
L
T N;
Rd
T Nb
%
T %
Y,
$=$
H, (z)
= +.,
RL -
-
c
aa=0
2 H(z)
@) --
Y-
bz
T N,’
Nbr
%
R,
Ni
RL -
b,-
_-
b ‘= aI
&
H, (2)
=
&
02
. H (2)
(4
FIG. 15. VII. Conclusions
A new wave digital filter has been proposed in this paper. Doubly terminated lossless ladder filters were considered where the different series and shunt arm elements or the different L-sections, etc., were characterized by transfer scattering matrices. Using bilinear transformation 8 = (z- l)/(z+
l),
each transfer scattering matrix was transformed to a corresponding chain matrix of a wave digital two-port and the individual digital two-ports were cascaded to realize the overall chain matrix. The two-port characterization of the analog circuit elements eliminates the need for “adapters” to interconnect the wave two-ports. For a given ladder analog filter, it has been shown that one can obtain eight digital realizations, which are all canonic with respect to both delays and multipliers. The sensitivity properties as well as the round off noise of these different realizations are now under study.
Vol.
300,
No.
1, July
1975
57
ill. N. S. Swamy ami K. S. Thyagarajan It has also been shown that by assuming all the port resistances of the ladder elements as two-ports to be equal, one can obtain a realization which is canonic with respect to multipliers only. This particular realization was found to be comparable with the one discussed in Ref. [2], from the point of view of sensitivity with respect to coefficient variations. It was also seen that this implementation was more desirable than the conventional cascade form as it requires smaller coefficient register lengths.
Acknowledgement This paper is based on a part of the doctoral dissertation to be submitted by K. S. Thyagarajan to Concordia University, Montreal, and was supported by the National Research Council of Canada under Grant A-7739.
References
(1) A. Fettweis, (2)
(3) (4)
(5) (6) (7) (8)
58
“Digital filter structures related to classical filter networks”, Arch. Elek. ubertragung, Band 25, pp. 79-89, Sept. 1971. K. Renner and S. C. Gupta, “On the design of wave digital filters with low sensitivity properties”, IEEE Trans. Circuit Theory, Vol. CT-20, pp. 555-567, Sept. 1973. A. Sedlmeyer and A. Fetteweis, “Digital filters with true ladder configuration”, Int. J. Circuit Theory Applications, Vol. 1, pp. B-10, March 1973. A. Sedlmeyer and A. Fetteweis, “Realization of digital filters with true ladder configuration”, Proc. IEEE Int. Symp. on Circuit Theomj, Toronto, pp. 149-152, 1973. interreciprocity and t,ransposition in wave digital A. Fetteweis, “Reciprocity, filters”, Ilzt. J. Circuit Theory Applications, Vol. 1, pp. 323-337, Dec. 1973. A. G. Constantinides, “Alternative approach to the design of wave digital filters”, Electron. Letta., Vol. 10, pp. 59-60, March 1974. B. Gold and C. M. Rader, “Digital Processing of Signals”, McGraw-Hill, New York, 1969. S. K. Mitra and R. S. Sherwood, “Digital ladder networks”, IEEE Trans. Audio & Electroacoust., Vol. AU-21, pp. 30-36, Feb. 1973.
Journal
of The Franklin
Institute
and K. s. THYAGARAJAN
Department of Electrical Engineering Concordia University *, Montreal, Canada
ABSTRACT
LC-ladder equivalent
:
This paper
proposes
networks by replacing digital two-port.
from doubly terminated amlEelement of the ladder by its
a new wave digital filter derived each series
or shunt
It is shown that such two-ports
may be cascaded without the use
of adapters defined by Fettweis
(1).A number of realizations of the wave dig&al two-ports, which are canonic with respect to both multipliers and delays, have been obtained. Also a realization
which
is canon&
with respect
considered
using this realization.
coeficient
change8 due to finite
to multipliers
The sensitivity
word length is compared
digital filter and also the one proposed
is given
and an example
with the conventional
by Renner and Gupta. It is
to be a more desirable form of implementation and cornparable to that of Renner and Gupta (2).
Jilter appears form
only
of this jilter with respect to the multiplier
found that the
than the conventional
cascaded proposed
cascade
I. Introduction
The design of digital filters which have low sensitivity to multiplier coefficient variations is currently receiving attention. Ideally it would be desirable to realize a digital filter whose transfer characteristics, namely the magnitude and phase responses, remain invariant to changes in the multiplier coefficients due to their finite word lengths. Under such situations, the length of the coefficient register can be kept to a minimum thus resulting in reduction in the cost of hardware units. To this end, Fettweis (1) has proposed a class of filters known as wave digital filters which exhibit much less variation in the transfer characteristics due to multiplier coefficient changes than the conventional form of digital filters do. Using Fettweis’ technique, Renner and Gupta (2) have given procedures for designing digital filters patterned after maximum available power (MAP) networks. These filters are, in general, non-canonio with respect to the multipliers. However, due to the low sensitivity properties, the lengths of the coefficient registers for the wave digital filters need be smaller than those of the conventional digital filters. This decrease in the coefficient register lengths in wave digital filters may not only result in a decrease in computational time between samples but also in a reduction in the overall cost of the hardware units. The design of wave digital filters proposed by Fettweis and others (l-5) is based on imitating certain doubly terminated LC-ladder structures. Instead of the voltages and currents, incident and reflected voltage or current waves are * Incorporating Sir George Williams University and Loyola of Montreal.
41
M.
N.
S.
Swamy and K. S. Tlzyagarajan
used as the input and output signals in wave digital filters. Using a transmission line transformation, each two-terminal element in the analog filter is transformed to a corresponding two-terminal wave digital element with the incident and reflected waves as digital input and output. However, direct interconnection of these wave digital elements is not possible in general due to incompatibility of the waves between the elements. So digital “adapters” are used between the elements so that the waves are made compatible between the different wave digital elements. The use of these adapters in wave digital filters, however, seems to make the design of such filters more complicated. This paper considers a different design approach to wave digital filters starting with doubly terminated LC-ladder networks, wherein the different circuit elements are represented as wave digital two-ports. It will be shown in a later section that the wave two-port characterization of the analog elements makes it possible to interconnect the different ports directly without the use of adapters. This makes the design procedure very simple. A number of different canonic realizations for the wave digital two-port is given. Also, an example is taken to exemplify the sensitivity properties of one of the realizations and those of the filters realized in the usual cascaded form or by Renner and Gupta’s method (2).
ZZ. Characterization
of the
Wave
Digital
Two-port
Consider Fig. 1, where the black box 8 may contain an interconnection of linear elements. If we now define the wave variables (&, gi), i = 1,2, by
(1) where R, and R, are arbitra,ry port resistances at ports 1 and 2, then we may express (& &) in terms of (&, a,) as (6)
where A+R,C+B+R,D p=---F-
2R,’
x = A+R,C 2
B+R,D -
2R,
’ >
fi=
R=
42
(3)
A-R&B-RID 2
2R,
A-R,C
B-RID
2
-
2R,
’
I
Jo:mal
of The
Franklin
Institute
A New Type of Wave Digital Filter
and chain matrix of m. II--c -..
-
1
(4) 12
LINEAR NETWORK
,
Rz
1
I
FIG. 1.
It may be observed that the determinant of [p] is
~F~=$(AD-BC). 2
In general, the elements of [p] are functions bilinear transformation (7) a=-
of S. If we now use the
z-1 z+l
and identify a”, and a”, with the input, and gl and g2 with the output variables of a digital two-port, then we may recognize (2) as the chain matrix description (8) of a digital two-port. We shall write (2) with (6) substituted as
Kj=[‘:11[I =[Fl[:j,
where
[F]=[P]
withs=-.
z-1
z+l
(74
Pb)
If N is reciprocal, it is seen from (5) and (7b) that IFI =$. 2
(8)
Thus, given any two-port network m with port resistances R, and R,, ale may describe a corresponding wave digital two-port N, whose chain matrix is given by (7). It is clear that if fi is a cascade of two-ports, say, fll, R2, . . . , iV,, then the corresponding digital two-port is the cascade of the digital two-ports N&N,, . . . . N, where Ni is the digital two-port corresponding to fli ; of course, it is assumed here that the output port resistance of fld is the same as the input port resistance Of lo,+l. ZZZ. Realization
of the Digital
Two-port
We shall now consider the realization of a digital two-port starting with a doubly terminated lossless network fl (Fig. 2), since it is known to have very
Vol.
300, No.
1, July
1975
43
M. N. S. Swamy and K. S. Thyagarajan
T (5) = V, /V, FIG. low
2.
sensitivities with respect to the different elements. It is seen from Fig. 2
that v, = v,-JIBR,,
(9)
v, = -12RRL.
(10)
Hence, from (1) and (lo), we have d, = J,$
(11)
and g, =
2Ji
(12)
l+d’
where +=-.
h--R, R,+R,
(13)
= (l+e)a,,
(14)
Also, from (1) and (9), we have a”,+&, where v, = CSi,,
(15) (16)
Hence, 6, 2% Ej=(l+fj)~=iq~*
2
v,
(17)
Also from (2) and (14), (p+Xl#&
= (l+e)G&--e(~+k~)9,
or
If we denote
v,
H(z) = v,
z-1
with s = z+l
(19)
then b 2 -?- -H(z), a, 1+$,
44
(204
Journal of The Franklin Institute
A New Type of Wave Digital Filter where
a2 =
WC)
+b,,
a, = (l+O)a,-Bb,. The corresponding realization two-port corresponding to fl.
Pod)
is shown in Fig. 3, where N is the digital
I=- RL - Re RL + Rz
it&= ?aI ‘+o
H (4
t
H (z) = -
V?.
With
I = g
VS
FIG. 3.
Thus, given the realization for T(s) = (I?&) as a lossless two-port can obtain the corresponding realization for H(z) = [V,/V,] with S = [(z-
l)/(z+
fl, we
l)]
by using Fig. 3, where the digital two-port N is obtained from the corresponding LC-network 8. Until now, we have not made any assumptions about 19. However, in order to realize N from &’ in a simple fashion, we shall assume fl to be a lossless ladder. Then the series elements can only be an L, C or an LC-parallel circuit, while the shunt elements are L, C or a series LC-circuit. Thus, it is necessary to derive the digital two-ports corresponding to only these elements, which we shall do in succeeding sections. Then, we may cascade them to form N and use Fig. 3 to realize H(z). IV.
Design for
of Digital
all the Series
Two-ports
Corresponding
and Shunt
Elements
to the Case
R, = R,
in m
Let a series element in fl be denoted by ZJs), while a shunt element by Y,(s). Let 2, and Y, represent the functions with s = (z- l)/(z+ 1). Then, for a series element Z,, the digital two-port description is
(21)
Vol.
300,
No.
1, July 1975
45
M. N. S. Swamy and K. S. Thyagarajan with 111 =
VI =
2, + (R, + R,) 2R, ’
(22)
2, + (4 - R,) 2R, ’
& = (I-PI),
‘Q = (I-“,),
while for a shunt element Yb (23) where
x
=
V2 =
(24)
Y2+(Gl-Gz) 2G,
2
’
(1 -_cL2), K2 = (1 -A,).
In the simple case when R, = R, = R (say), (21) and (23) may be rearranged as
b, =
I
a,+~(al---a,),
b, = al-:(a,--a,) for the series element, and
(25)
I
b, = a,-~(a,+a,),
(26) b, = a,-~(aI+a,)
I
for the shunt element. For illustration, the corresponding realizations are shown for series L and shunt C in Fig. 4(a) and (b). Realizations for series C and shunt L may be obtained by changing z-l to -z-l and L to (l/C) in Fig. 4(a) and (b) respectively. Similarly, realizations may also be obtained for a parallel LC in the series arm and a series LC in the shunt arm. It should be pointed out that there are no delay-free loops in any of these realizations. However, when these are cascaded, delay-free loops appear, a situation which is similar to the one encountered in wave digital filters discussed in Refs. (1) and (2). In order to overcome this difficulty, we introduce an extra delay element either in the forward path or in the return path, where two sections are cascaded. It should be pointed out that one can always choose either R = R, or R = R, in Fig. 2, so that, in Fig. 3 either $ = 0 or 0 = 0.
46
Journal of The Franklin
Institute
A New Type
of Wave
Digital
Filter
A design example which is considered in Ref. (2) will now be taken. Let it be required to design a third-order low pass Butterworth filter with a cut-off frequency of 100 Hz and sampling frequency of 10 kHz. Since
R
@
!/L +_
:
c
+
_t
2-l
I
_+
@ R
I
oa2 <=
(2R-L)/(2R+L) (a)
b,
0
02
a=(ZG-C)/(ZG+C)
(b) FIG.
bilinear transformation prewarped using
is being
4.
used,
the
critical
w, = tan(+,T),
frequency
must
be (27)
where wA and wD correspond to the critical frequencies of the analog and digital filters, respectively. The analog filter is shown in Fig. 5 with the
“S:l”aTG--Jj
T(4= v,/ V, r WA= 0.031426266 FIG. 5. corresponding digital filter in Fig. 6. In order to examine the effect of quantization of the multiplier coefficients, plots of the magnitude response of the wave digital filter of the above example are shown in Fig. 7 for different levels of fixed point quantization. To make a valid comparison the same digital transfer function is implemented in the conventional cascade form
Vol. 300, No. 1, July 1975
47,
M. N. 8. Swamy and K. S. Tiiyagarajan as shown in Fig. 8, and its corresponding plots are shown in Fig. 9. From the two plots, we observe that the shift in the magnitude response due to coefficient changes is less for the proposed wave digital filter than that for the conventional cascade filter. It is found that for this example, realization is possible even with word lengths of 5 bits for multipliers for the proposed
da = -0.88172593,
4
= -0.939062506,
FIG.
ds = 0;
6.
0
-1 -1.5 * .s
-2
f f
-2.5
p -3 t
-3.5 -4 -4.5 -5 0
10
20
30
40
50
60
70
80
90
100
110
-_,H.Z
FIG.
D(l+z
4
-=
01
( I + AZ-’
-1 )3
) ( I + Bz-’
FIG.
48
7.
+ CZ’~
)
8.
Journalof The FranklinInstitute
A New Type of Wave Digital Filter wave digital filter while a minimum of 15 bits are required for the conventional cascade filter and 10 bits for the filter proposed by Renner and Gupta (2). The different parameters of interest are listed in Table I. From the entries in 0.5 0 -0.5
-1 -1.5 B .5 -2 0 3 -2.5 .z P E f
-3 -3.5 -4 -4.5 -5
I
I
0
10
20
30
40
50
FIG.
60
70
80
90
1M) *HZ
110
9.
TABLE I Cascaded $.?ter L 3 dB frequency (Hz) o/o error Gain at 0 Hz (dB)
15
16
1’7
18
89.5 - 10.5 0.425
100 0 0.391
99.8 - 0.2 0.409
99.9 -0.1 0.400
19 100 0 -0.16
00 100 0 0
Renner and Gupta’s jilter y 3 dB frequency (Hz) o/o error Gain at 0 Hz (dB)
15
16
17
18
19
co
99.8 -0.2 - 0.0668
100.2 0.2 - 0.0053
99.9 -0.1 - 0.0293
100 0 0.005
100 0 0.005
100 0 0
Proposed filter y
5
3 dB frequency (Hz) o/o error Gein at 0 Hz (dB) % error = f+.
m
106.8 6.8 5.7
15 99.9 -0.1 -0.052
16 100 0 0.0151
17 100 0 -0.0174
18 100 0 -0.00118
19 100 0 -0.000917
co 100 0 0
100.
Vol. 300, No. 1, July 1975
49
M. N. S. Swamy and K. S. Thyagarajan the table, we observe that the proposed wave digital filter is more desirable than the conventional cascade implementation. It may also be noted that the proposed wave digital filter is comparable to that of Renner and Gupta.
V.
Canonic
realization
of Wave
Digital
Two-ports
Until now, it had been assumed that the port resistances R, and R, were equal for each of the individual two-ports in m. Under this condition, even though the corresponding digital two-ports had no delay-free loops, cascade of such two-ports had delay-free loops. In order to remove these loops, we had to introduce extra delays. We shall now show that by relaxing the condition R, = R,, we can obtain digital filters which are canonic with respect to both delays and multipliers. The digital two-port descriptions corresponding to series and shunt elements are respectively given by (21) and (23) and may be rewritten as 6, = a2 + (4~~) (a, - a2), b2 = a2 + (I/A
(a, - a2)
1
for the series element, and 6, = a2 s + (44
(a, + u2a,),
b2 = - a2 + (I/p2) (a1 + o2 a,),
(Jo= 4/R,
I
(29)
for the shunt element. These may be realized as shown in Fig. 10(a) and (b). It is seen from these figures that there are delay-free paths from a2 to b, in
2
(4
@) FIG. 10.
the realization of both the series and shunt elements. Thus, if (vJpl) and (v2/p2) are made to have no delay-free paths, then there will be no delay-free paths from a, to b1 ; as a consequence, there will be no delay-free loops when two sections are cascaded. In order to ensure this, we should make sure that the numerators of (vl/pl) and (v2/p2) should be of at least one degree lower in z than their corresponding denominators. We shall first show how we can achieve this for a series L. For this element, we may write
z(R,-R,+L)+(R,-RI-L) ;I = z(R,+R,+L)+(R,+R,-L)’
50
(30)
Journal of The Franklin
Institute
A New Type
of Wave
Digital
Filter
To ensure (vr/pr) to have no delay-free paths, we make (R,-R,+L)
= 0
In this case, the digital two-port
or
R,
description
= (R,+L).
(31)
becomes
(32) b, = a,+o
where
R,
R,
(33)
a=iTl=(Rz+
The realization for (32) is shown in Table II. Table II also shows realizations when the series element is a C or an LC-parallel circuit. TABLE
First
set of realizations
corresponding
II
to the series from a, to b,
SAME AS NETWORK
N, WITH
Z
element
REPLACED
2,
BY (-Z-
w&h no delay-free
puth
)
We shall now consider the shunt element; the chain matrix [FP] of the corresponding digital two-port ND is given by (23). Let N,’ represent a digital two-port, whose transfer matrix is the transpose of that of N; such a two-port can be obtained from N by simply reversing the arrows in N. Then the chain matrix of N,’ is given by
[F,l, =
$[“”“1, v2
Vol.300,No. 1,July1975
(344
K2
51
M. N. S. Swamy and K. S. Thyagarajan where A=]F,]=$?
(34b)
2
If we now identify Ya with Z,, G, with R, and G, with R,, we see that (34) reduces to --A, [-r(‘,]k = P1 . (35) [ -Q Kl I The realization for the above is the same as that of 2, with b, changed to -b, and a2 to -a2. Hence, the realization for the shunt element Yb may be obtained by starting with the realization of the series element for which 2, = Y6, transposing it, changing b, to - b,, a2 to - a2 and replacing R, by G, and R, by G,. Thus, for a shunt capacitor C, the realization is as shown in Table III, where u = (G2/G1), G, = G,+ C. The realizations for an L and a TABLE III
First set of realizations ELEMENT
corresponding to the shunt element Yz with no delay-free path from a, to b, CORRESPONDING
WAVE
NETWORK SAME AS NETWORK
N4 WITH
DIGITAL
RELATIONS
2 -PORT
??
N ‘, Z -’
NETWORK
REPLACED
N5
BY ( -2-l
d,Gr
)
GI G, = Gz + I a-=
GZ r !
d=
I - LC ItC
G, = Gz + ,&
0
I NETWORK
N6
series LC circuit in the shunt arm are shown in Table III. It should be observed that there is no delay-free path from a, to b, for any of these realizations. Thus, any of the networks N,-N may be cascaded without delay-free loops. It is also seen that if networks N,-N, are used, we may always choose the port resistance R, of the last element in the lossless network to be load resistance RL. We shall refer to the corresponding digital realization as realization I(a). We may obtain an alternate realization for the series element (and hence for the shunt element), and thus obtain an alternate realization for the
52
Journal of The Franklin Institute
A New Type of Wave Digital Filter digital filter. Consider the series element for which the chain matrix [F] of the corresponding digital two-port N, is given by (21). Let [Fr] be the chain matrix of the two-port obtained by reversing the ports of N,. Then, (36) From (21) and (36), it is seen that [F,] = [F] if we interchange the roles of R, and R,. Thus, a second realization for 2, can be obtained by simply reversing the ports of the realizations given in Table II and interchanging R, and R, in those realizations. These are shown in Table IV. Similarly, a TABLE IV
Second
set of realizations
corresponding to the series from a2 to b, CORRESPONDING
ELEMENT
WAVE DIGITAL
element
2 - PORT
2,
with no delay-free
path
RELATIONS
g=- RI b ;gi-Jy]~g
A: $ a
R2 = R, + L
0
6=-
RI R2
F? 0
R2 = R,
+-
0
Rz = R,+--I 1 LC
second set of realizations corresponding to YZ may be obtained using Table III and are shown in Table V. It is seen that when these networks are cascaded, there would be no delay-free loops. Further, if these are used in obtaining N for Fig. 3, we may always assume that the port resistance R, of the first element in the lossless network 8 is equal to R, in Fig. 2. We shall refer to the corresponding digital realization as realization II(a). For the sake of illustration, consider the network of Fig. 11. Then, using networks N,-N, we may obtain realizations I(a) and II(a); these are shown in Figs. 12 and 13. In Fig. 12, the different networks are N,:
network N,
with a = G,lG,,
N, :
network N1
with a = RJR,
NC:
network N4
with CJ= G,/G,
Nd:
network Ns
with u = RJR,,
Vol. 309, No. 1, July 1975
ar= (1 - L4 C,)/( 1 i- L, C,), (37) 01= (1 - L, C,)/( 1 + L,C,)
53
M. N. 5. Swamy
and K. S. Thyagarajan TABLE V
Second set of realizations corresponding to the shunt from a2 to b,
I
ELEMENT
CORRESPONDING
element Yz with no delay-free path
WAVE DIGITAL
2 - FORT
G2 =G,+C
.j= -G, G.2
G2 =G,++
G2
c = GI + I+L~,
I
I
FIG. 11. Nd
Rc
Nc
_^
Rb
Nb
R,
N,
G,
C4 = GL + I + LqC*
I$ = R, + L,
G,
= Gb + Cz
%=R=+j&
@=Rd-R, Rd + R,
$=
H, (L)
RL -0
_c
bz
cIz=o
a 1 = 2 H (z)
FIG. 12. bz 0 :
bz-
-
RA 9-
NA
RB
NB
RC
NC
RD
L
GC=GD+Ca
I
+ +,
Rb=RC+Ls +=$$&
RS
___a===%
R~ ‘Rs
ND
01
GA=
, ,
:=
C4 GB + I + L4 c4
Ha(z)
=++
H(z)
FIG. 13.
54
Journalof
The Franklin
Institute
A New Type of Wave Digital Filter and the function realized is b 2 = H,(z) = 2H(z), a,
(38)
where z-l
H(z) = 3
with s = -
Zfl
8
.
(39)
In Fig. 13, the different networks are N,:
network
hT, with a = R,/R,,
N,:
network N4
with (T= CD/G,,
NB:
network Nr
with (5 = RJR,,
N,:
network N6
with c = GBIGA,
and the corresponding
a! = (l-L,CJ(l
+L,C,), (40)
(II= (1 - L4C4)/( 1 + L,C,)
I
function realized is b -2 = H,(z) = -%
H(z)
(4la)
l+d
a1
with
4= VI. Alternate
Canonic
R,-RA
(41b)
R,+R,’
Realizations
We have shown in the previous section that we can always get two canonic realizations for the digital filter of Fig. 3, starting with the analog network of Fig. 2. In both these realizations, N is a cascade of subnetworks whose chain matrices are of the form (7) with (F 1= (RJR,). We have termed these realizations I(a) or II(a) depending on whether Tables II and III or IV and V have been used. We shall now show that we can always get three other realizations corresponding to each of I(a) and II(a). We call these realizations (b), (c) and (d), respectively. Realization (b) Let the realization (a) [I(a) or II(a)] be as in Fig. 3. Let the chain matrix of N be [F] as given by (7). Hence, we may express [;:I
(42a)
= q]
where R,P
[F,l=Iz;
[
_)I
-v
K * I
(42b)
If we now put the constraints
Vol. 300,
No.1,July
1975
a, = - Ob,,
(434
a2 = (I-+)a,++b,,
(43b)
55
M. N. S. Swamy and I<. S. Thyagarajan then
Using (13), (20a) and (2Ob), we may reduce the above to
The corresponding
b R 1 -1=1 RL r8 zH(z). as realization is shown in Fig. 14.
b, _=-. 0s
RI RL
is*
(44)
2 H(z)
FIG. 14.
Realization (c) Let NT be the digital two-port whose transfer matrix is the transpose of that of N. This may be obtained from N by simply reversing the direction of the arrows in N. Then the chain matrix of NT is
where A = IF]. If [F] is given by (7), then
[F,] =
2f [
h
I.
(45b)
K
Thus, if N is replaced by NT in Fig. 3, we see that b R 1+e AC-2 R,p+h$+~B+~hj as or
(46)
This will be referred to as realization
(c) for H(z).
Realization (d) Similarly, it may be shown that if N is replaced by NT in Fig. 14, we get b = R 2 A 2 R, G-i3 H(zL as which we will refer to as realization (d) for H(z).
56
(47)
Journal of The
Franklin
Institute
A New Type of Wave Digital Filter As an example, let us obtain realizations I(b), I(c) and I(d) from the realization I(a) given in Fig. 12. These are shown in Fig. 15(a), (b) and (c). Similarly, realizations II(b), II(c) and II(d) may be obtained from II(a) of Fig. 13. a,-
-
b,
__
Rd Nd R,
NC
Nb
lb
b
R~
Na
02
A
b,
-= 01
Rd -*RL
1. 1+e
R,
N,
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Rd
T Nb
%
T %
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$=$
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-
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FIG. 15. VII. Conclusions
A new wave digital filter has been proposed in this paper. Doubly terminated lossless ladder filters were considered where the different series and shunt arm elements or the different L-sections, etc., were characterized by transfer scattering matrices. Using bilinear transformation 8 = (z- l)/(z+
l),
each transfer scattering matrix was transformed to a corresponding chain matrix of a wave digital two-port and the individual digital two-ports were cascaded to realize the overall chain matrix. The two-port characterization of the analog circuit elements eliminates the need for “adapters” to interconnect the wave two-ports. For a given ladder analog filter, it has been shown that one can obtain eight digital realizations, which are all canonic with respect to both delays and multipliers. The sensitivity properties as well as the round off noise of these different realizations are now under study.
Vol.
300,
No.
1, July
1975
57
ill. N. S. Swamy ami K. S. Thyagarajan It has also been shown that by assuming all the port resistances of the ladder elements as two-ports to be equal, one can obtain a realization which is canonic with respect to multipliers only. This particular realization was found to be comparable with the one discussed in Ref. [2], from the point of view of sensitivity with respect to coefficient variations. It was also seen that this implementation was more desirable than the conventional cascade form as it requires smaller coefficient register lengths.
Acknowledgement This paper is based on a part of the doctoral dissertation to be submitted by K. S. Thyagarajan to Concordia University, Montreal, and was supported by the National Research Council of Canada under Grant A-7739.
References
(1) A. Fettweis, (2)
(3) (4)
(5) (6) (7) (8)
58
“Digital filter structures related to classical filter networks”, Arch. Elek. ubertragung, Band 25, pp. 79-89, Sept. 1971. K. Renner and S. C. Gupta, “On the design of wave digital filters with low sensitivity properties”, IEEE Trans. Circuit Theory, Vol. CT-20, pp. 555-567, Sept. 1973. A. Sedlmeyer and A. Fetteweis, “Digital filters with true ladder configuration”, Int. J. Circuit Theory Applications, Vol. 1, pp. B-10, March 1973. A. Sedlmeyer and A. Fetteweis, “Realization of digital filters with true ladder configuration”, Proc. IEEE Int. Symp. on Circuit Theomj, Toronto, pp. 149-152, 1973. interreciprocity and t,ransposition in wave digital A. Fetteweis, “Reciprocity, filters”, Ilzt. J. Circuit Theory Applications, Vol. 1, pp. 323-337, Dec. 1973. A. G. Constantinides, “Alternative approach to the design of wave digital filters”, Electron. Letta., Vol. 10, pp. 59-60, March 1974. B. Gold and C. M. Rader, “Digital Processing of Signals”, McGraw-Hill, New York, 1969. S. K. Mitra and R. S. Sherwood, “Digital ladder networks”, IEEE Trans. Audio & Electroacoust., Vol. AU-21, pp. 30-36, Feb. 1973.
Journal
of The Franklin
Institute