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A special class of multiplierless digital filters
A Special Class of Multiplierless Digita/ Filters by STAVROS \'.AROUF.AKIS Computer Center, Athens, Greece
Nuclear
Research
Center Democritos,
Aghia Paraskeui,
and AN.ASTAS[OSVENETSANOPOULOS Department of Electrical Canada MSS lA4
Engineering,
University
of
Toronto,
Toronto,
ABSTRACT. This paper de& with rhe design of discrete low-puss filters. wing a special c/ass of polynomials of the second and third order. \b.ifh coeficients rc+ich are powers of two. To take adcantage of the characteristics of these polynomials, the discrete The filters so derived are multiplierless integrator is used as a basic component. resulting in hardware economy. Simulation indicates that the filters obtained are equally good or better thun those obtained by the application of the impulse invariance method.
Nomenclature original analog transfer function normalized analog transfer function H(z) digital transfer function F,(z) transfer function resulting from f,(z) transform F?(z) transfer function resulting from fz(z) transform transfer function resulting from f,(z) transform K(z) G(z) transfer function resulting from impulse invariance method original cut-off frequency WO new cut-off frequency W, sampling period TS coefficient of denominator polynomial b, constant which is a power of two digitizing transform function f(: transform constant C complex variables in the analog domain 3, P complex variable in the digital domain coefficients using impulse invariance method PI, El denominator numerator coefficient using impulse invariance method imaginary part of s-domain root ;: (T real part of s-domain root. H(P)
H,(s)
0
The
Franklin
lnstiruk
0016-M)3?/8?11?0333-l~~3.0010
333
S. Varoufakis
and A. Venetsanopoulos
I. Introduction
The s-domain design of analog filters, using integrators as basic components, has the advantage that high-Q poles may be achieved with low sensitivity to amplifier change of gain (1.2). Other successful approaches simulate the inductance by differentiators and thus achieve a close analog to the passive RLC ladder structure (3). It is also possible to realize a transfer function H(s) on an analog computer (Fig. la), using only integrators, adders and coefficient multipliers (4,5). This realization is very attractive, because its elements may be easily replaced by the corresponding digital elements, such as digital integrators (6), adders and multipliers. Since multipliers are more costly than adders, it is of interest to consider a special class of filters that does not require multiplications in their realization. The purpose of this paper is to investigate the existence of polynomials, of the second and third order, that possess coefficients which are powers of two. These polynomials may be used to realize digital filters. The filters thus derived are multiplierless and have small realization error and hardware economy.
II. Reduction of Polynomial Coeficients
The discrete simulation of the integrators, for filter synthesis, combines the s-domain characteristics with digital flexibility. s-domain properties of transfer functions are well established and known. Consider the all-pole transfer function H(p)
=
A.
(1)
2 b’ i=O For an appropriate choice of pole locations, this function may represent an analog low-pass filter. If the coefficients of (1) are positive and powers of two, the realization through discrete integrators is both economical and simple. This is due to the error-free multiplications, which can be implemented by a shift of the multiplied word. In practice, the determination of such coefficients must be carried out so that (1) satisfies the following minimum constraints: (i) The magnitude response of the resulting filter must be similar to that of the original filter, and (ii) stability must be preserved. These constraints were investigated for second and third order polynomials by using software simulation techniques. The results obtained are presented in Tables I and II. The polynomials were obtained under the following two restrictions: (i) The maximum attenuation ripple in the pass-band was required not to exceed the value of t 3 dB. 334
Journal
oi the Franklin Institute Tcrgamon Press Ltd.
,*I
Gzz
G
Group
I 4 16 64
I
I I
32
8
2
1 I
I
I 2 4 8 2 4 8
2 4 8
8
2 4
Coefficients b, b,
I 4 I6 64 I I I
1 1 I
2 8 32
b”
H(P)
=
0.86600
0.50000 I .OOOOO 2.oootnl 4.00000 0.25000 0.12500 0.06250
bd + b,p + bu
11”
3.46410 6.92820 0.43301 0.21650 0.10820
1.73200
0.25000 0. I2500
0.50000 4.00000
I .ooooo 2.00000
0.25000 0.12500
0.50000 4.00000
I .ooooo 2.00000
Roots Negative real part ImFpart
H,,(S)=
I .2720 2.5440 5.088 I 10.1762 0.6360 0.3180 0.1590
0.3536 0.1768
0.707 I 5.6569
h/d
K(s)
(011
(Jo
b b,, bL.sL+~.s+~
- 3db
1.4142 2.8284
lff(ju,,)~~
- ,sz + , .414\.~+ ,
Normalized polynomial s = i’/W,,
- 12.60
- 9.29
M;tximurn roll-off db/oct;ive
;; i
rz ‘2ii
?) 5’ $T
% WIs 7” $7 $5 ar
o\
w
w
G r33
Gr32
G r31
Group
2 4
1
4
4 I6
x 64
I 2
I I
I
16
I6
4 8
1 I
4
8 32
8 64
2
8
I
I
b,
b,
4 8
32
2 8
4
2
4 I6
4 8
32
2 8
b,
Coefficients
I-l(P)
I I
64
I 8
1
I
4 32
1 t
64
1 8
b,
=
b,p’
+ b,p:
0.28492 0.14246
2.27936
0.56984 1.13968
0.78260
I .56520
3.13040 6.26079
0.50000 0.25000
h + b,p + b,,
0.05377
0.36032 0. IO754
0.21508 0.430 I6
0.10870
0.2 I740
0.43480 0.86960
0.25000 0. I2500
2.00000
1.oOOOO
4.oOwO
0.50000
1.ooooo
Imaginary Negative real root
2.00000
Negative real root
Third order polynomials
H.,(s)
0.32679
5.22857 0.65357
1.30714 2.6 1420
0.26086
0.52171
I .04343 2.08685
0.43301 0.21651
3.46410
0.86603 I .73205
Im. :art
roots
=
powers
w
b 1s~+!+_~y+h,,
2. wo
Do/w:,
0.38 I2
6.0988 0.7624
I .5247 3.0494
0.3753
0.7505
1.5010 3.002 I
0.5000 0.2500
4.0000
2.0000
of two
- 3 db
1.oooo
]H(jw,,)rz
with coeficients
TAULE II
=
I.183
w:
H.,(s) = I s’ 0.656s’ + + 0.860s + 0.282
I.183 s3-t2.665s2t1.775s+
H.(s)
Normalized polynomial s = p/w,,
- 24.50
-- I.93
- 15.12
db/oct;lve
Wll-Off
Maximum
A Special Class of Multiplierless Digital Filters (ii) The roll-off frequency was limited to be no less than the corresponding roll-off of a Butterworth of the same degree. All possible combinations of coefficients less or equal to 64 were considered, for all possible second and third order polynomials. At the same time we determined the polynomial cut-off frequency wO, which is also shown in Tables I and II. This search can be easily extended to higher order filters (7) and longer coefficients and transfer functions with numerator polynomials. The normalization of the polynomials, with respect to the cut-off frequency oo, leads to the formation of two groups for the second and three groups for the third order polynomials, respectively. The first group in each table corresponds to an exact Butterworth polynomial, while the other groups are Chebyshev-like types. The normalization is obtained through the linear substitution, p = was
(2)
where the s stands for the normalized, while p stands for the original complex frequency. Introducing (2) in (1) yields the normalized analog transfer function H”(S)
=
,~
bo’w:
(3)
2 bisiIw;-i 1=0
III.
Conversion
Starting introduce
of the Analog Filter to a Digital Filter
from the analog transfer function H,,(s) given a new cut-off frequency wc, which results in (4)
by (3), we may
(4)
This equation may be digitized, if we choose a convenient transform of the form s +(c/T,)f(z), for a differentiator or s-‘+(T,/c)f-l(z), for an integrator and replace the corresponding analog components by digital components in the simulation diagram of the analog filter. Different forms for the function f(z) have been considered in the literature (6) so that the resulting filter is realizable. Such a substitution results in the following transfer function:
(5)
where period
T, is the integrator and c is a constant,
Vol. 314. No. 6. pp. 333-346. December Prinled in Great Britain
time constant, which where c E 1,2 (6).
corresponds
to the sampling
1981
337
S. Varoufakis
and A. Venetsanopoulos
For a given polynomial and transform there are two variables in(j), the cut-off frequency of the analog filter w,. and the sampling period T,, These two variables should be chosen, so that the cut-off frequency of the digital filter is controlled and the term in the parenthesis of (5) becomes a coefficient, which is a power of two. This implies that
(6) where M is a constant, obtain
which is a power of two. Introducing
(6) in (_5), we
f(z) should not be substituted here by the bilinear transform, because it leads to delay-free loops and therefore unrealizable filters. It is possible, however, to use the bilinear transform, perform the necessary operations to express the resulting transfer function as a rational function of z, and implement the filter in a direct way. The filter will then have coefficients, which are expressed as a sum of powers of two. In the following we shall apply three types of discrete integrators and study the corresponding magnitude responses for a specific set of coefficients. The results thus obtained will be compared to the results obtained by application of the impulse invariance method. Note that in Fig. lb, the multiplier Is/c, which is part of the integrator’s structure, is incorporated in the filter’s coefficients. Thus the derived second order ail pole low-pass filter is a discrete counterpart version of the analog filter. Transform of the form f,(z) = z- I An application may be illustrated for the case of N = 2 and for the second set of coefficients of the G,zz group in Table I, where bO= 4 b, = 2 bz = 1
and
wo=2.5440.
A simulation diagram for the corresponding analog filter is shown in Fig. la. The simple integration form, known as the direct z-transform method or DDI (6) may be used to realize a low-pass transfer function. The corresponding transform is obtained for c = 1 and f,(z) = z - 1, where the integrator transform is s-l--, T,(l/(z - l))(J, of Fig. lb), and boM-* F’(z) = (b2 - b,M-’ + boM-I) - (2 - b,M-‘)z’+
The resulting multiplierless 338
b$’
digital filter is shown in Fig. lc. Journal
of the Franklin Instirurs Pergamon Press Ltd.
A Special Class of Multiplierless Digital Filters
i___----___----__-----__---___-_--_____~ (b)
n
n
FIG. 1. Second order low-pass filter transfer function realization. (a) With analog integrators, (b) direct-transform digital integrator DDI, (c) corresponding transformed circuit with DDI.
It should be noted that the transform in use, which replaces the first derivative by a forward difference, does not satisfy the requirement of mapping the jw axis to the unit circle of the z-domain and as a result stability is only preserved when the poles of the resulting filter lie within the unit circle Consider
the roots
of a second-order s = -u00,+
The resulting
poles
denominator j&0,;
~~20.
polynomial (9)
are (10)
Using
(6), we obtain *=1_c!?E!!?+jc__.L!%~
Al-The stability
constraint
is given
R M
(11)
by the inequality
(12) Vol. 314. No. 6. pp. 333-346. Printed in Great Britain
December
198?
339
S. Varoufakis and A. Venetsanopoulos
which is equivalent
to (13)
For the third-order results in
Rearranging
polynomial,
the real root must also be checked.
This
(13) and (14) and using (6), we obtain
(1% and f5 2 +.
(16)
Thus the sampling frequency fs depends on the roots of the normalized polynomial and on the analog cut-off frequency f,. For the third-order polynomial (13) and (14) determine the minimum value of M, in order to preserve stability. It is evident that we have to choose the largest of the two values of M. The minimum M and f, are evaluated for the second polynomial of the G,?? group to obtain bo=4
b1 =2
bz = 1 M = 4 fs ~9.879Hz.
(17)
Table III contains the reduced transfer functions, representing the five different groups of polynomials of Tables I and II. Returning to (8) and introducing the coefficients of (17), we obtain 4iW’
F,(z) = (1 - 2&f-’ + 4K?) - 2( 1 - M-‘)z + z2*
(18)
For different values of M, M = 4,8,16 . . . and for the cut-off frequency of the digital filter being 1000 Hz, plots may be obtained for the magnitude response (Fig. 2). The passband overshoot is due to the mapping of the jo axis to a line tangential to the unit circle in the z-domain (8). Transform of the form f*(z) = (z - 1)/z”’ (6)
Another simple transform, the “lossless discrete integrator”, has been proposed in the literature, which may be used to realize (7). Consider s -+ (c/T,)(z - 1)/z”‘, which for z = e’“r~results in j(2/7’,) sin (wT,/2), (c = 1). 340
Journal ofrhe Franklin
lnst~tute Pergamon Press Ltd.
A Special Class of Multiplierless
Digital Filters
TABLE III Low-pass transferfunctions calculated for f,(z) where j stands for the transform in use, j = 1, 2, 3
Reduced transfer function Hz,(z) =
H::(z) =
for A4 = 4
1
.0.25000 + jO.2500
1 + 4f;(z) + 8fi(z) 1 1 + 4f,(z) + 16f;(z)
- 0.12500
1 H31(z) = 1 + 8f;( z) f 32f;(z) + 64f;( z) H,J z) =
2
jO.216506
- 0.2500 - 0.125 k j0.2 16506 - 0.782599
1
1 +4f,(z)+ H-i,(z) =
Values of f,(z) which made the denominator zero
16f;(z)+
16f;(z)
1 1 f 8f,(z) + 16f;(z) + 64f;(z)
- 0.1087 c jO.260856 - 0.14246 - 0.0537699 t 1’0.326785
2
0
-2
P -4 0 c -6 : 3 Z C -8 z I -10
-12 f-Hz -14
100
250
400
550
700
850
loco
1150
13ca
1450
1600
FIG. 2. Magnitude response of a second order low-pass transfer function when f,(z) = z - 1 transform is applied. Vol. 314. No. 6. pp. 333-346. December Printed in Great Britain
1982
341
S. Varoufakis and A. Venetsanopoulos
For the integrator (J! of Fig. lb), S -’
-3
T,z-“2/(l
- *-I).
(19)
Introducing this transform in (7), we obtain the corresponding transfer function F?(z). In contrast to the previous transform, a better bandpass magnitude response is obtained in Fig. 3, for the G,?? coefficients, although the sampling frequency applied is low (M = 1). This transform has to be applied with caution because of stability and realization problems, due to the half powers of z-‘. When the filter is found unstable it should be stabilized by pole inversion. This may result in the loss of the powers of two property of the coefficients. Transform of the form f3(z) = (z - l)z/(z + 1)
This transform is the modified bilinear transform, in which the delay-free forward path is eliminated by introducing a delay. As a consequence, the mapping of the s-plane jw axis onto the unit circle in the z-plane does not occur. Care must be taken in choosing M to preserve stability and stabilization, by pole inversion, may have to be applied in this case also. Introducing f3(z) in (7) and rearranging the coefficients, we obtain F,(z) = 1+2z+z2 boM-‘boM-2
+ (2boW2 - b,M-‘)z
+ (boM-’ + bJz2 + (b,M-’
-2b2)r3 + b&’ (20)
Figure 4 shows the corresponding magnitude responses for M = 8 and M = 16, which approximate closely the corresponding analog transfer function. The
-75
-9.0
1 -
+ \
-105
f--HI
-12.0
. 100
I
250
LOU
550
7C-J
850
lOCII
1150
,300
FIG. 3. Magnitude response of a second order low-pass transfer f?(z) = (z - I)/z”’ transform is applied.
342
1‘50
1600
function
when
Joun~al of the
Franklin Institute Pergamon Press Ltd.
A Special frequency introduced Impulse
Class
distortion due to the transform by appropriate choice of w,. invariance
of Multiplierless
Digital Filters
was high and a compensation
method
To evaluate the results the transfer function of a analog transfer function, analog normalized transfer
obtained by these transforms we may now obtain second order digital filter, starting from the same but applying the impulse invariance method. The function is 0.618
(21)
H,(s) = 0.618 + 0.786s + s’ which is derived Once the method
was
from the normalization is applied it yields G(Z)
of the polynomial
previously
used.
‘l’-’
=
1 - p,z-* + pzz-? where
the calculated
coefficients
77 = 0.296812,
are
p1 = 1.413836
and
pZ = 0.606587.
The magnitude response of (22) is plotted in Fig. 5, for infinite (floating point) and for finite word-length coefficients of 4 bits.
word-length We observe
-12.
-14
-
100
250
FIG. 4. Magnitude
4co
sso
700
850
,000
1150
1x0
response of a second order low-pass transfer f3(z) = (z - l)z/(z + 1) transform is applied.
Vol. 314. No. 6. pp. 333-346. December Printed in Grear Britain
1450
Jeoo
function
when
1981
343
S. Varoufakis
and A. Venetsanopoulos
that the resulting response derived from the coefficients’ finite word-length truncation, is different from that obtained using floating point arithmetic for the coefficients. Table IV compares the hardware requirements of these realizations. It is clear that the first filters based upon f,(z) and f3(z) transforms result in hardware economy. More generally, it can be stated that for a specific cut-off frequency, it is possible to obtain two multiplierless filters with a second order transfer function and three from the third order transfer function. These filters have good properties and can be used in those applications where hardware economy and accuracy are at a premium and where frequency responses, similar to those of the proposed filters, are adequate. This is of course accomplished at the expense
4.
/G(Jw)/ 2.
/G(jw)
-2.
I 4
Bits
%
-12: ‘-I* -14
1 100
FIG. 5.
260
400
550
a50
700
loco
1302
11%
1450
1600
Comparison of the magnitude response for infinite word-length (floating point), IG(jw)l and finite word-length coefficients of 4 bits.
TABLE IV Comparative
Transform
344
table for hardware realization requirements of two different types of jilters
used
Registers
Adders
Multipliers
f,(z) f3(z)
2 4
4 6
0 0
Impulse invariance method
6
2
3
Journal of the
Franklin Institute Pergamon Press Ltd.
A Special
Class
of Multiplierless
Digital Filters
of higher sampling frequencies. The family of the proposed filters is shown in Table III. The idea developed in this paper can be extended to realize two-dimensional recursive filters (9). These filters are easily derived through a transformation, which may be successfully applied to the five groups obtained here. A similar difference routing digital filter (DRDF), especially suited for application in data transmission, was reported in the literature (10). IV. Summary
and Conclusions
This paper considers the use of the discrete integrator, as basic block for filter synthesis, in order to take advantage of the characteristics of some polynomials, with coefficients which are powers of two. Its main features may be outlined as follows: (i) Generation of special polynomials, with coefficients which are powers of two; (ii) Use of the discrete integrator, as the basic element for filter synthesis, in conjuction with the polynomials developed. A second-order polynomial was considered as an example. The following transform methods were applied: (a) (b) (c) (d)
The The The The
f,(z) = f?(z) = f3(z) = inpulse
I - 1 transform; (-_ - 1)/z’ ’ transform; (z - l)r/(z + 1) transform; invariance method.
The results obtained indicate that the first transform exhibits an overshoot in the passband. The third transform results in a response which, for a higher sampling frequency (M = 16), is close to that obtained by the analog transfer function. The second and third transforms increase the order of the filter and may result in unstable designs, which should be stabilized. The absence of multipliers reduces time and hardware and contributes in the minimization of the errors produced. The advantage of the multiplierless operation renders the proposed transfer functions useful for digital filter implementation with microprocessors. A known drawback of the discrete integrator is the high sampling frequency required. However, the multiplierless discrete filter can be operated with reasonably low sampling frequency. The results obtained are compared to those obtained by the filters derived from the application of the impulse invariance method and indicate that the filters proposed possess simpler hardware structure and characteristics equivalent or better to those obtained by the impulse invariance method. From the simulation results it is clear that the transforms in use introduce a frequency distortion that must be tolerated or compensated if a response similar to that of the analog must be achieved. Such a distortion is also present in the impulse invariance method with finite word length. To this end and for purposes of comparison of the different curves to the initial one lH(jw)(, this distortion was compensated in order to achieve -3 dB droop at f = 1000 Hz. Vol. 314. No. 6. pp. 333-346. Lkember Printed in Great Britain
1982
345
S. Varoufakis
and A. Venetsanopoulos
References (1) J. Tow, “Design formulas for active RC filters using operational amplifier biquad”, Eiectr. Left., Vol. j, pp. 339-341, 1969. (2) L. Thomas, “The biquad: Part I-Some practical design considerations”, IEEE Trans. Circuit Theory, Vol. CT-18, pp. 350-357, 1971. (3) H. J. Orchard, “Inductorless filters”, Efectr. Lett., Vol. 2, pp. 224-225, 1963. (4) L. P. Huelsman, “Active Filters: Lumped, Distributed, Integrated, Digital and Parametric”, pp. j2-60, McGraw-Hill, New York, 1979. (5) S. K. Mitra, “Analysis and Synthesis of Linear Active Networks”, pp. 489-492, John Wiley, New York, 1969. (6) L. T. Bruton, “Low-sensitivity digital ladder filter”, IEEE Trans. Circuits Systems, Vol. CAS-22, pp. 168-176, 1975. (7) S. J. Varoufakis, “Multiplierless digital differential filters”, First Int. Conf. on Sciences Systems, pp. 826830, Greece, August 1976. (8) L. R. Rabiner and B. Gold, “Theory and application of digital signal processing”, pp. 21&215, Prentice-Hall, Englewood Cliffs, New Jersey, 1975. (9) J. M. Costa and A. N. Venetsanopoulos, “Design of circularly symmetric twodimensional recursive filters”, IEEE Trans. Acoustics Speech and Signal Processing, Vol. ASSP-22, pp. 432-443, 1974. (10) P. J. Gerwen, W. F. G. Mecklenbrauker, N. A.M. Verhoeckx, F. A. M. Snijders and H. A. Van Essen, “A new type of digital filter for data transmission”, IEEE Trans. Communications, Vol. COM-23, pp. 222-234, 1973.
346
Journal
of the Franklin Institute Pergamon Press Ltd.
Nuclear
Research
Center Democritos,
Aghia Paraskeui,
and AN.ASTAS[OSVENETSANOPOULOS Department of Electrical Canada MSS lA4
Engineering,
University
of
Toronto,
Toronto,
ABSTRACT. This paper de& with rhe design of discrete low-puss filters. wing a special c/ass of polynomials of the second and third order. \b.ifh coeficients rc+ich are powers of two. To take adcantage of the characteristics of these polynomials, the discrete The filters so derived are multiplierless integrator is used as a basic component. resulting in hardware economy. Simulation indicates that the filters obtained are equally good or better thun those obtained by the application of the impulse invariance method.
Nomenclature original analog transfer function normalized analog transfer function H(z) digital transfer function F,(z) transfer function resulting from f,(z) transform F?(z) transfer function resulting from fz(z) transform transfer function resulting from f,(z) transform K(z) G(z) transfer function resulting from impulse invariance method original cut-off frequency WO new cut-off frequency W, sampling period TS coefficient of denominator polynomial b, constant which is a power of two digitizing transform function f(: transform constant C complex variables in the analog domain 3, P complex variable in the digital domain coefficients using impulse invariance method PI, El denominator numerator coefficient using impulse invariance method imaginary part of s-domain root ;: (T real part of s-domain root. H(P)
H,(s)
0
The
Franklin
lnstiruk
0016-M)3?/8?11?0333-l~~3.0010
333
S. Varoufakis
and A. Venetsanopoulos
I. Introduction
The s-domain design of analog filters, using integrators as basic components, has the advantage that high-Q poles may be achieved with low sensitivity to amplifier change of gain (1.2). Other successful approaches simulate the inductance by differentiators and thus achieve a close analog to the passive RLC ladder structure (3). It is also possible to realize a transfer function H(s) on an analog computer (Fig. la), using only integrators, adders and coefficient multipliers (4,5). This realization is very attractive, because its elements may be easily replaced by the corresponding digital elements, such as digital integrators (6), adders and multipliers. Since multipliers are more costly than adders, it is of interest to consider a special class of filters that does not require multiplications in their realization. The purpose of this paper is to investigate the existence of polynomials, of the second and third order, that possess coefficients which are powers of two. These polynomials may be used to realize digital filters. The filters thus derived are multiplierless and have small realization error and hardware economy.
II. Reduction of Polynomial Coeficients
The discrete simulation of the integrators, for filter synthesis, combines the s-domain characteristics with digital flexibility. s-domain properties of transfer functions are well established and known. Consider the all-pole transfer function H(p)
=
A.
(1)
2 b’ i=O For an appropriate choice of pole locations, this function may represent an analog low-pass filter. If the coefficients of (1) are positive and powers of two, the realization through discrete integrators is both economical and simple. This is due to the error-free multiplications, which can be implemented by a shift of the multiplied word. In practice, the determination of such coefficients must be carried out so that (1) satisfies the following minimum constraints: (i) The magnitude response of the resulting filter must be similar to that of the original filter, and (ii) stability must be preserved. These constraints were investigated for second and third order polynomials by using software simulation techniques. The results obtained are presented in Tables I and II. The polynomials were obtained under the following two restrictions: (i) The maximum attenuation ripple in the pass-band was required not to exceed the value of t 3 dB. 334
Journal
oi the Franklin Institute Tcrgamon Press Ltd.
,*I
Gzz
G
Group
I 4 16 64
I
I I
32
8
2
1 I
I
I 2 4 8 2 4 8
2 4 8
8
2 4
Coefficients b, b,
I 4 I6 64 I I I
1 1 I
2 8 32
b”
H(P)
=
0.86600
0.50000 I .OOOOO 2.oootnl 4.00000 0.25000 0.12500 0.06250
bd + b,p + bu
11”
3.46410 6.92820 0.43301 0.21650 0.10820
1.73200
0.25000 0. I2500
0.50000 4.00000
I .ooooo 2.00000
0.25000 0.12500
0.50000 4.00000
I .ooooo 2.00000
Roots Negative real part ImFpart
H,,(S)=
I .2720 2.5440 5.088 I 10.1762 0.6360 0.3180 0.1590
0.3536 0.1768
0.707 I 5.6569
h/d
K(s)
(011
(Jo
b b,, bL.sL+~.s+~
- 3db
1.4142 2.8284
lff(ju,,)~~
- ,sz + , .414\.~+ ,
Normalized polynomial s = i’/W,,
- 12.60
- 9.29
M;tximurn roll-off db/oct;ive
;; i
rz ‘2ii
?) 5’ $T
% WIs 7” $7 $5 ar
o\
w
w
G r33
Gr32
G r31
Group
2 4
1
4
4 I6
x 64
I 2
I I
I
16
I6
4 8
1 I
4
8 32
8 64
2
8
I
I
b,
b,
4 8
32
2 8
4
2
4 I6
4 8
32
2 8
b,
Coefficients
I-l(P)
I I
64
I 8
1
I
4 32
1 t
64
1 8
b,
=
b,p’
+ b,p:
0.28492 0.14246
2.27936
0.56984 1.13968
0.78260
I .56520
3.13040 6.26079
0.50000 0.25000
h + b,p + b,,
0.05377
0.36032 0. IO754
0.21508 0.430 I6
0.10870
0.2 I740
0.43480 0.86960
0.25000 0. I2500
2.00000
1.oOOOO
4.oOwO
0.50000
1.ooooo
Imaginary Negative real root
2.00000
Negative real root
Third order polynomials
H.,(s)
0.32679
5.22857 0.65357
1.30714 2.6 1420
0.26086
0.52171
I .04343 2.08685
0.43301 0.21651
3.46410
0.86603 I .73205
Im. :art
roots
=
powers
w
b 1s~+!+_~y+h,,
2. wo
Do/w:,
0.38 I2
6.0988 0.7624
I .5247 3.0494
0.3753
0.7505
1.5010 3.002 I
0.5000 0.2500
4.0000
2.0000
of two
- 3 db
1.oooo
]H(jw,,)rz
with coeficients
TAULE II
=
I.183
w:
H.,(s) = I s’ 0.656s’ + + 0.860s + 0.282
I.183 s3-t2.665s2t1.775s+
H.(s)
Normalized polynomial s = p/w,,
- 24.50
-- I.93
- 15.12
db/oct;lve
Wll-Off
Maximum
A Special Class of Multiplierless Digital Filters (ii) The roll-off frequency was limited to be no less than the corresponding roll-off of a Butterworth of the same degree. All possible combinations of coefficients less or equal to 64 were considered, for all possible second and third order polynomials. At the same time we determined the polynomial cut-off frequency wO, which is also shown in Tables I and II. This search can be easily extended to higher order filters (7) and longer coefficients and transfer functions with numerator polynomials. The normalization of the polynomials, with respect to the cut-off frequency oo, leads to the formation of two groups for the second and three groups for the third order polynomials, respectively. The first group in each table corresponds to an exact Butterworth polynomial, while the other groups are Chebyshev-like types. The normalization is obtained through the linear substitution, p = was
(2)
where the s stands for the normalized, while p stands for the original complex frequency. Introducing (2) in (1) yields the normalized analog transfer function H”(S)
=
,~
bo’w:
(3)
2 bisiIw;-i 1=0
III.
Conversion
Starting introduce
of the Analog Filter to a Digital Filter
from the analog transfer function H,,(s) given a new cut-off frequency wc, which results in (4)
by (3), we may
(4)
This equation may be digitized, if we choose a convenient transform of the form s +(c/T,)f(z), for a differentiator or s-‘+(T,/c)f-l(z), for an integrator and replace the corresponding analog components by digital components in the simulation diagram of the analog filter. Different forms for the function f(z) have been considered in the literature (6) so that the resulting filter is realizable. Such a substitution results in the following transfer function:
(5)
where period
T, is the integrator and c is a constant,
Vol. 314. No. 6. pp. 333-346. December Prinled in Great Britain
time constant, which where c E 1,2 (6).
corresponds
to the sampling
1981
337
S. Varoufakis
and A. Venetsanopoulos
For a given polynomial and transform there are two variables in(j), the cut-off frequency of the analog filter w,. and the sampling period T,, These two variables should be chosen, so that the cut-off frequency of the digital filter is controlled and the term in the parenthesis of (5) becomes a coefficient, which is a power of two. This implies that
(6) where M is a constant, obtain
which is a power of two. Introducing
(6) in (_5), we
f(z) should not be substituted here by the bilinear transform, because it leads to delay-free loops and therefore unrealizable filters. It is possible, however, to use the bilinear transform, perform the necessary operations to express the resulting transfer function as a rational function of z, and implement the filter in a direct way. The filter will then have coefficients, which are expressed as a sum of powers of two. In the following we shall apply three types of discrete integrators and study the corresponding magnitude responses for a specific set of coefficients. The results thus obtained will be compared to the results obtained by application of the impulse invariance method. Note that in Fig. lb, the multiplier Is/c, which is part of the integrator’s structure, is incorporated in the filter’s coefficients. Thus the derived second order ail pole low-pass filter is a discrete counterpart version of the analog filter. Transform of the form f,(z) = z- I An application may be illustrated for the case of N = 2 and for the second set of coefficients of the G,zz group in Table I, where bO= 4 b, = 2 bz = 1
and
wo=2.5440.
A simulation diagram for the corresponding analog filter is shown in Fig. la. The simple integration form, known as the direct z-transform method or DDI (6) may be used to realize a low-pass transfer function. The corresponding transform is obtained for c = 1 and f,(z) = z - 1, where the integrator transform is s-l--, T,(l/(z - l))(J, of Fig. lb), and boM-* F’(z) = (b2 - b,M-’ + boM-I) - (2 - b,M-‘)z’+
The resulting multiplierless 338
b$’
digital filter is shown in Fig. lc. Journal
of the Franklin Instirurs Pergamon Press Ltd.
A Special Class of Multiplierless Digital Filters
i___----___----__-----__---___-_--_____~ (b)
n
n
FIG. 1. Second order low-pass filter transfer function realization. (a) With analog integrators, (b) direct-transform digital integrator DDI, (c) corresponding transformed circuit with DDI.
It should be noted that the transform in use, which replaces the first derivative by a forward difference, does not satisfy the requirement of mapping the jw axis to the unit circle of the z-domain and as a result stability is only preserved when the poles of the resulting filter lie within the unit circle Consider
the roots
of a second-order s = -u00,+
The resulting
poles
denominator j&0,;
~~20.
polynomial (9)
are (10)
Using
(6), we obtain *=1_c!?E!!?+jc__.L!%~
Al-The stability
constraint
is given
R M
(11)
by the inequality
(12) Vol. 314. No. 6. pp. 333-346. Printed in Great Britain
December
198?
339
S. Varoufakis and A. Venetsanopoulos
which is equivalent
to (13)
For the third-order results in
Rearranging
polynomial,
the real root must also be checked.
This
(13) and (14) and using (6), we obtain
(1% and f5 2 +.
(16)
Thus the sampling frequency fs depends on the roots of the normalized polynomial and on the analog cut-off frequency f,. For the third-order polynomial (13) and (14) determine the minimum value of M, in order to preserve stability. It is evident that we have to choose the largest of the two values of M. The minimum M and f, are evaluated for the second polynomial of the G,?? group to obtain bo=4
b1 =2
bz = 1 M = 4 fs ~9.879Hz.
(17)
Table III contains the reduced transfer functions, representing the five different groups of polynomials of Tables I and II. Returning to (8) and introducing the coefficients of (17), we obtain 4iW’
F,(z) = (1 - 2&f-’ + 4K?) - 2( 1 - M-‘)z + z2*
(18)
For different values of M, M = 4,8,16 . . . and for the cut-off frequency of the digital filter being 1000 Hz, plots may be obtained for the magnitude response (Fig. 2). The passband overshoot is due to the mapping of the jo axis to a line tangential to the unit circle in the z-domain (8). Transform of the form f*(z) = (z - 1)/z”’ (6)
Another simple transform, the “lossless discrete integrator”, has been proposed in the literature, which may be used to realize (7). Consider s -+ (c/T,)(z - 1)/z”‘, which for z = e’“r~results in j(2/7’,) sin (wT,/2), (c = 1). 340
Journal ofrhe Franklin
lnst~tute Pergamon Press Ltd.
A Special Class of Multiplierless
Digital Filters
TABLE III Low-pass transferfunctions calculated for f,(z) where j stands for the transform in use, j = 1, 2, 3
Reduced transfer function Hz,(z) =
H::(z) =
for A4 = 4
1
.0.25000 + jO.2500
1 + 4f;(z) + 8fi(z) 1 1 + 4f,(z) + 16f;(z)
- 0.12500
1 H31(z) = 1 + 8f;( z) f 32f;(z) + 64f;( z) H,J z) =
2
jO.216506
- 0.2500 - 0.125 k j0.2 16506 - 0.782599
1
1 +4f,(z)+ H-i,(z) =
Values of f,(z) which made the denominator zero
16f;(z)+
16f;(z)
1 1 f 8f,(z) + 16f;(z) + 64f;(z)
- 0.1087 c jO.260856 - 0.14246 - 0.0537699 t 1’0.326785
2
0
-2
P -4 0 c -6 : 3 Z C -8 z I -10
-12 f-Hz -14
100
250
400
550
700
850
loco
1150
13ca
1450
1600
FIG. 2. Magnitude response of a second order low-pass transfer function when f,(z) = z - 1 transform is applied. Vol. 314. No. 6. pp. 333-346. December Printed in Great Britain
1982
341
S. Varoufakis and A. Venetsanopoulos
For the integrator (J! of Fig. lb), S -’
-3
T,z-“2/(l
- *-I).
(19)
Introducing this transform in (7), we obtain the corresponding transfer function F?(z). In contrast to the previous transform, a better bandpass magnitude response is obtained in Fig. 3, for the G,?? coefficients, although the sampling frequency applied is low (M = 1). This transform has to be applied with caution because of stability and realization problems, due to the half powers of z-‘. When the filter is found unstable it should be stabilized by pole inversion. This may result in the loss of the powers of two property of the coefficients. Transform of the form f3(z) = (z - l)z/(z + 1)
This transform is the modified bilinear transform, in which the delay-free forward path is eliminated by introducing a delay. As a consequence, the mapping of the s-plane jw axis onto the unit circle in the z-plane does not occur. Care must be taken in choosing M to preserve stability and stabilization, by pole inversion, may have to be applied in this case also. Introducing f3(z) in (7) and rearranging the coefficients, we obtain F,(z) = 1+2z+z2 boM-‘boM-2
+ (2boW2 - b,M-‘)z
+ (boM-’ + bJz2 + (b,M-’
-2b2)r3 + b&’ (20)
Figure 4 shows the corresponding magnitude responses for M = 8 and M = 16, which approximate closely the corresponding analog transfer function. The
-75
-9.0
1 -
+ \
-105
f--HI
-12.0
. 100
I
250
LOU
550
7C-J
850
lOCII
1150
,300
FIG. 3. Magnitude response of a second order low-pass transfer f?(z) = (z - I)/z”’ transform is applied.
342
1‘50
1600
function
when
Joun~al of the
Franklin Institute Pergamon Press Ltd.
A Special frequency introduced Impulse
Class
distortion due to the transform by appropriate choice of w,. invariance
of Multiplierless
Digital Filters
was high and a compensation
method
To evaluate the results the transfer function of a analog transfer function, analog normalized transfer
obtained by these transforms we may now obtain second order digital filter, starting from the same but applying the impulse invariance method. The function is 0.618
(21)
H,(s) = 0.618 + 0.786s + s’ which is derived Once the method
was
from the normalization is applied it yields G(Z)
of the polynomial
previously
used.
‘l’-’
=
1 - p,z-* + pzz-? where
the calculated
coefficients
77 = 0.296812,
are
p1 = 1.413836
and
pZ = 0.606587.
The magnitude response of (22) is plotted in Fig. 5, for infinite (floating point) and for finite word-length coefficients of 4 bits.
word-length We observe
-12.
-14
-
100
250
FIG. 4. Magnitude
4co
sso
700
850
,000
1150
1x0
response of a second order low-pass transfer f3(z) = (z - l)z/(z + 1) transform is applied.
Vol. 314. No. 6. pp. 333-346. December Printed in Grear Britain
1450
Jeoo
function
when
1981
343
S. Varoufakis
and A. Venetsanopoulos
that the resulting response derived from the coefficients’ finite word-length truncation, is different from that obtained using floating point arithmetic for the coefficients. Table IV compares the hardware requirements of these realizations. It is clear that the first filters based upon f,(z) and f3(z) transforms result in hardware economy. More generally, it can be stated that for a specific cut-off frequency, it is possible to obtain two multiplierless filters with a second order transfer function and three from the third order transfer function. These filters have good properties and can be used in those applications where hardware economy and accuracy are at a premium and where frequency responses, similar to those of the proposed filters, are adequate. This is of course accomplished at the expense
4.
/G(Jw)/ 2.
/G(jw)
-2.
I 4
Bits
%
-12: ‘-I* -14
1 100
FIG. 5.
260
400
550
a50
700
loco
1302
11%
1450
1600
Comparison of the magnitude response for infinite word-length (floating point), IG(jw)l and finite word-length coefficients of 4 bits.
TABLE IV Comparative
Transform
344
table for hardware realization requirements of two different types of jilters
used
Registers
Adders
Multipliers
f,(z) f3(z)
2 4
4 6
0 0
Impulse invariance method
6
2
3
Journal of the
Franklin Institute Pergamon Press Ltd.
A Special
Class
of Multiplierless
Digital Filters
of higher sampling frequencies. The family of the proposed filters is shown in Table III. The idea developed in this paper can be extended to realize two-dimensional recursive filters (9). These filters are easily derived through a transformation, which may be successfully applied to the five groups obtained here. A similar difference routing digital filter (DRDF), especially suited for application in data transmission, was reported in the literature (10). IV. Summary
and Conclusions
This paper considers the use of the discrete integrator, as basic block for filter synthesis, in order to take advantage of the characteristics of some polynomials, with coefficients which are powers of two. Its main features may be outlined as follows: (i) Generation of special polynomials, with coefficients which are powers of two; (ii) Use of the discrete integrator, as the basic element for filter synthesis, in conjuction with the polynomials developed. A second-order polynomial was considered as an example. The following transform methods were applied: (a) (b) (c) (d)
The The The The
f,(z) = f?(z) = f3(z) = inpulse
I - 1 transform; (-_ - 1)/z’ ’ transform; (z - l)r/(z + 1) transform; invariance method.
The results obtained indicate that the first transform exhibits an overshoot in the passband. The third transform results in a response which, for a higher sampling frequency (M = 16), is close to that obtained by the analog transfer function. The second and third transforms increase the order of the filter and may result in unstable designs, which should be stabilized. The absence of multipliers reduces time and hardware and contributes in the minimization of the errors produced. The advantage of the multiplierless operation renders the proposed transfer functions useful for digital filter implementation with microprocessors. A known drawback of the discrete integrator is the high sampling frequency required. However, the multiplierless discrete filter can be operated with reasonably low sampling frequency. The results obtained are compared to those obtained by the filters derived from the application of the impulse invariance method and indicate that the filters proposed possess simpler hardware structure and characteristics equivalent or better to those obtained by the impulse invariance method. From the simulation results it is clear that the transforms in use introduce a frequency distortion that must be tolerated or compensated if a response similar to that of the analog must be achieved. Such a distortion is also present in the impulse invariance method with finite word length. To this end and for purposes of comparison of the different curves to the initial one lH(jw)(, this distortion was compensated in order to achieve -3 dB droop at f = 1000 Hz. Vol. 314. No. 6. pp. 333-346. Lkember Printed in Great Britain
1982
345
S. Varoufakis
and A. Venetsanopoulos
References (1) J. Tow, “Design formulas for active RC filters using operational amplifier biquad”, Eiectr. Left., Vol. j, pp. 339-341, 1969. (2) L. Thomas, “The biquad: Part I-Some practical design considerations”, IEEE Trans. Circuit Theory, Vol. CT-18, pp. 350-357, 1971. (3) H. J. Orchard, “Inductorless filters”, Efectr. Lett., Vol. 2, pp. 224-225, 1963. (4) L. P. Huelsman, “Active Filters: Lumped, Distributed, Integrated, Digital and Parametric”, pp. j2-60, McGraw-Hill, New York, 1979. (5) S. K. Mitra, “Analysis and Synthesis of Linear Active Networks”, pp. 489-492, John Wiley, New York, 1969. (6) L. T. Bruton, “Low-sensitivity digital ladder filter”, IEEE Trans. Circuits Systems, Vol. CAS-22, pp. 168-176, 1975. (7) S. J. Varoufakis, “Multiplierless digital differential filters”, First Int. Conf. on Sciences Systems, pp. 826830, Greece, August 1976. (8) L. R. Rabiner and B. Gold, “Theory and application of digital signal processing”, pp. 21&215, Prentice-Hall, Englewood Cliffs, New Jersey, 1975. (9) J. M. Costa and A. N. Venetsanopoulos, “Design of circularly symmetric twodimensional recursive filters”, IEEE Trans. Acoustics Speech and Signal Processing, Vol. ASSP-22, pp. 432-443, 1974. (10) P. J. Gerwen, W. F. G. Mecklenbrauker, N. A.M. Verhoeckx, F. A. M. Snijders and H. A. Van Essen, “A new type of digital filter for data transmission”, IEEE Trans. Communications, Vol. COM-23, pp. 222-234, 1973.
346
Journal
of the Franklin Institute Pergamon Press Ltd.