ARC High Order Filters Suitable for Antialiasing and/or
Reconstruction Filters
Abstract – The discrete time signal processing circuit requires an anti-aliasing filter at the input and a
reconstruction filter at the output, generally. In this paper, selected basic structures of some active filters are
described and compared with a view to the degradation of the attenuation over the transient frequency of the
operational amplifier. Firstly, the reasons for the degradation of the attenuation are explained theoretically.
Secondly, these conclusions are verified by simulations. These simulations were performed by spice-like circuit
simulator MicroCap version 11.
Keywords- low-pass filter structure; real operational amplifier; frequency response; decreasing of the attenuation
at the high frequencies; transient frequency of the operational amplifier
Received: June 22, 2021. Revised: March 17, 2022. Accepted: April 20, 2022. Published: May 18, 2022.
1. Introduction
HE main disadvantage of the all discrete-time
signal processing circuits is the periodicity of
its frequency response, as is depicted in Fig.1.
Therefore, the main task of the antialiasing filter at
the input is the attenuation of frequencies upper half
of sampling frequency [1], [2], [3]. The realization
of these filters can use active RC filters as well,
where an active element can be an operational
amplifier, i.e. filter in the voltage mode.
Figure 1. Digital filter attenuation.
In the following text, the use of the real
operational amplifier in basic filter structures will
be discussed.
Some biquad filters structures are characterized
by a degradation of the attenuation at high
frequencies [4], [5]. This degradation of the
attenuation occurs only for some filters of the even
orders, i.e. for the biquads, and is caused by the
final value of the output resistance of the used
operational amplifier [6] (see Fig. 2).
Figure 2. A frequency response of the low-pass Sallen_Key
structury biquad
The reason of these attenuation losses (i.e. for ideal
case) for a second-order low-pass filter is depicted
in Fig. 2 as well. As is shown, the ideal course of
the second-order low-pass filter frequency response
is a monotonous decreasing of the magnitude value
over the cutoff frequency 0 at whole frequency
fSA f(Hz)
L
(dB)
fSA/2fCfS
+
_
R
R
C
2.C
+
_
VA.V
R
R
R
R
R
R
rO
rO
F()
0 dB/dec
-40 dB/dec
+20 dB/dec
0 dB/dec
0
T
0
-20log 1
2+
A
rI
R
rO
ideal case
influence of
finite T
BOHUMIL BRTNÍK
University of Pardubice
náměstí Čs.Legií 565, 586 01 Pardubice
CZECH REPUBLIC
T
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.10
Bohumil Brtník
E-ISSN: 2224-266X
94
range with –40 dB/dec slope. But the finite value of
the transient frequency T of the used real
operational amplifier leads to the break of this ideal
slope (i.e. –40 dB/dec). The result is a loss of
attenuation in the stopband for the frequencies
around the transient frequency of the used
operational amplifier T.
Filter characteristics in the transition area
between pass-band and stop-band frequencies are
described in the generally available literature
frequency [2], [3], [4] and many more.
For example, there are two variants possible,
when Huelsman low-pass structure filter is
designed by comparison with the general relation
for low-pass transfer (1).
2
O
O
2
2
O
4354535251
31
ω
Q
ω
ss
ω
YYYYYYYYYY
YY
(1)
Behind the input resistor R, in the node 2
generally either a grounded capacitor C2 and a
feedback resistor R3 or a grounded resistor and a
feedback capacitor can be connected (where C2 and
R3 are changed). However, the first variant is
always chosen, with a grounded capacitor (see
Fig.3).
Figure 3. Structure of the LP filter in first variant
In this case the other part of this filter is thus
excited by the zero voltage. Then even the output
voltage at high frequencies must be zero, as is
required for an ideal low pass filter. Then the input
and grounded resistor create a voltage divider,
which proceeds to the output of the biquad via the
feedback capacitor. As a result, with a non-zero
output resistance r0 of the operational amplifier (see
Fig.2), the output voltage for the high frequency
from filter is non zero, as well.
2. LP-Filters Structure
In this paragraph, the basic low-pass filter type
structures will be critically evaluated from the point
of view described above. At the same time, the
circuit diagram of the filter will always be replaced
by the circuit valid for frequencies above the transit
frequency of the operational amplifier.
2.1 Filter with distributed feedback loop
These filters using only one operational amplifier
as a voltage follower exhibit the lowest known
sensitivity Q to the passive elements. The general
filter structure is described in [13], for 3rd order is
shown in Fig. 4.
Figure 4. Structure of the LP filter with distributed feedback
loop
Consider that the amplification of an operational
amplifier is reduced to zero value at highest
frequencies. The reactance of all capacitors is at
highest frequencies equal to zero as well. In this
case can be drawn model of this circuit, which is
depicted in Fig.5.
Figure 5. Model of the LP filter with distributed feedback loop
in high frequencies
Consider for example
RRRRR 4321
, thus
if
i
r
is the input resistance and
o
r
the output
resistance of real operational amplifier, the output
voltage
O
V
is determined as (2).
0
33
3
33
3
!
OiOi
Oi
OiOi
Oi
IO
rrr
R
r
R
rr
R
R
rrr
R
r
R
rr
R
VV
(2)
Because normally ri is in hundreds of kiloohms, R
in kiloohms and ro in ohms, therefore in all cases
can be written
O
rR 
,
Oi rr 
, thus eq.(2) can be
simplified in following form (3)
O
O
I
O
rR
r
V
V
(3)
_
+
R1R4
C2
14
2 3
R3C5
_
+
R1R3
R4
C2VO
C3
C4
R2
C1
VI
R1R3R4VO
R2
VI
rI
rO
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.10
Bohumil Brtník
E-ISSN: 2224-266X
95
Consider now commonly filter value R= 15 kOhm.
For commonly used operational amplifier of type
LM741, which has following typical main
parameters ROUTAC = rO = 50 Ohm, A = 200 V/mV,
GBW = 1 MHz, we see that
5015000
50
202020 log
rR
r
log
V
V
log
O
O
I
O
dB),(,log 50482200033020
(4)
which is a typical value for this filter structure as
well.
2.2 LP Sallen-Key Structure of Even
Order
The Sallen-Key low-pass filter i.e. LP-SK [7]
using an operational amplifier as a voltage follower
exhibit the lowest sensitivity Q to the passive
elements. Therefore, from the viewpoint the
sensitivity of the filter, the composition from two
(generally even) biquads LP-SK is the best.
Commonly used high orders filter structure is
shown in Fig.6.
Figure 6. The LP filter 4th order SK structure
The main disadvantage of LP-SK biquad filters
for odd orders, however, is a reduction of the
attenuation in the stop-band. We know that the
amplification factor A of an operational amplifier is
reduced to zero at high frequencies. The reactance
of capacitors nears to zero for frequencies close to
infinite. The voltage ratio has a nonzero value in the
zone over of transient frequency of the operational
amplifier [9], this fact explains the degradation of
the filter properties. Consider now
RRRRR 4321
, thus output voltage
V
of
the first biquad (see. Fig.7) derived from the model
of the LP filter 4th order SK structure in high
frequencies, which is depicted in Fig.6 and is given
as Eq.(5)
Figure 7. Model of the LP filter 4th order SK structure in high
frequencies
0
!
oioi
oi
oioi
oi
I
r.rr.Rr.R
r.r.R
R
r.rr.Rr.R
r.r.R
.VV
(5)
Because in all typical cases (as is described above)
is
O
rR 
,
Oi rr 
, thus Eq.(5) can be simplified
as the first biquad voltage transfer in form (6)
o
o
IrR
r
V
V
(6)
Because the second one biquad voltage transfer is
the same, thus the resulting transfer of the LP filter
4th order SK structure in high frequencies is given
in Eq.(7)
0
2
2
)rR(
r
rR
r
rR
r
V
V
o
o
o
o
o
o
I
O
(7)
Consider again commonly filter value R= 15 kOhm.
For commonly used operational amplifier of type
LM741 with following parameters ROUTAC = rO =
= 50 Ohm, A = 200 V/mV, GBW = 1 MHz, thus we
see that
2
2
2
2
15050
50
202020 log
)rR(
r
log
V
V
log
O
O
I
O
dB),(,log 10096420101120 5
(8)
The approximately value -100 dB is in very well
correlation with simulation result by program
MicroCap version 11 which verifies this calculated
solution as is depicted in Fig. 8, where is the
resulting magnitude characteristics graph.
Figure 8. Magnitude characteristic of LP SK 4th order filer.
The result of the simulation is in full agreement
with the theoretical analysis, as follows from the
comparison of Fig. 8 and Fig.2.
2.3 Non-Cascade Structure
The non-cascading filter structure with galvanic
connection between the input Vi and the output Vo
voltage [10] is used when the transfer DC
component is required. A resistor Ro is connected
between input and output nodes, the frequency
variable impedance is realized by a circuit
containing one or more operational amplifiers,
_
+
R1R3R4
C2
VO
_
+
C3
C4
R2
C1
VI
R1R3R4VO
R2
VI
rI1
rO1
rI2
rO2
V
100
1K
10K
100K
1M
10M
-150.00
-120.00
-90.00
-60.00
-30.00
0.00
10.00
dB(v(10))
F (Hz)
Micro-Cap 10 Evaluation Version
circuitSK_SK_obec.cir
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.10
Bohumil Brtník
E-ISSN: 2224-266X
96
which are connected between the output node and
ground. The schematic diagram of the 4th order non-
cascading filter is depicted in following Fig. 9.
Figure 9. Schematic diagram of the LP filter 4th order non-
cascade structure
Figure 10 is a simplified schematic diagram of
this biquad at high frequencies (where all capacitors
are substituted to a short circuit and the operational
amplifier does not voltage amplify, in other word its
voltage amplify A=0).
Figure 10. Model of the LP filter 4th order non-cascade structure
in high frequencies
In all cases typically are
O
rR 
,
Oi rr 
, thus
output voltage
O
V
can be calculated in simplified
form (9)
oo
o
I
O
rR
r
V
V
(9)
For commonly filter members value R = 15
kOhm and operational amplifier LM741 with
following typical parameters ROUTAC = rO
= 50 Ohm, A = 200, GBW = 1 MHz, we see that
the damping above the transient frequency is
approximately -50 dB again (see calculation of the
same Eq.4 in the previous paragraph B).
2.4 Huelsmann Structure
Second one filter if the transfer DC component
is required is 4th order filter Huelsmann Structure
MFB. Its schematic diagram is depicted in Fig. 11,
where we can see galvanic connection between the
input and the output node (over the resistors R1, R3,
R4, R6).
Figure 11. LP biquad 4th order Huelsmann (MFB)
structure
Figure 12 is a simplified schematic diagram of the
circuit from Fig.11 at higher frequencies (where all
capacitors are substituted to a short circuit in the
case of ideal passive elements).
Figure 12. Model of the LP biquad 4th order Huelsmann (MFB)
structure
The voltage V = 0 i.e. the other part of this filter
is thus excited by the zero voltage. As a result, the
output voltage Vo must be equal to zero as well.
Therefore the magnitude of the voltage transfer ratio
at the highest frequencies Finf must be equal to zero,
too (10)
0 I
O
inf V
V
limF
(10)
where: Finf, VI, VO were specified above, thus the
damping above the transient frequency is infinite in
this case (11).
 )
V
V
(loglim
V
V
log
I
O
I
O
20
(11)
Note that this Huelsmann structure can be used
when the transfer DC component of the signal is
required.
2.5 2T Filter Structure
Now in this paragraph, we focus on the 2T LP
filter [7], [8] at the highest frequencies, as well.
Figure 13 is a schematic diagram of a 4th order filter,
which is able transferred DC component as well,
because there is a galvanic connection between the
input and the output over all resistors R.
_
+
R1
R0
C2VO
C3C4
R2
C1
VI
_
+
R3
R1
R0
VO
R2
VI
R3
rI1
rI2
rO1
rO2
R1
R3
R4
C2
VO
_
+C3
C4
R2
C1
VI
_
+
R6R5
R1
R3
R4VO
R2
VI
R6R5
rI1 rO1 V = 0 rI2 rO2
V = 0
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.10
Bohumil Brtník
E-ISSN: 2224-266X
97
Figure 13. LP biquad 4th order in 2T structure.
Figure 14 depicted a simplified schematic
diagram of the biquads from Fig.13 at higher
frequencies (where all capacitors C are simply
substituted to a short circuit).
Figure 14. Model LP biquad 4th order 2T structure.
The operational amplifier loses its amplification
factor, now. The voltage at the second, fourth, sixth
and seventh node, i.e. in four nodes is equal to zero
V=0, therefore the following parts of the filter is
excited by zero voltage, thus output voltage Vo must
be equal to zero, too. The magnitude of the voltage
transfer ratio at the highest frequencies must be
equal to zero as well (12)
0 I
O
inf V
V
limF
(12)
where: Finf, VI, VO were specified above, thus the
damping above the transient frequency is infinite as
well (see eq.(11)).
3. Discussion
Since the main function of both the anti-aliasing
filter in discrete-time signal processing (see Fig. 15)
is to suppress the frequency higher than half the
sampling frequency [11], [12], it is possible, based
on the theoretical analysis mentioned in paragraph
II, select appropriate filter structures, taking into
account the properties of real operational amplifier.
Figure 15. Antialiasing filter in signal processing.
In general, if the antialiasing filter is designed, it is
necessary consider following basic values: the
attenuations Acorner, Astop and the frequencies fcorner,
fstop for the reference LP filter. But, as is described
above, an important parameter is the attenuation
over the transient frequency Aover, as well. The
comparison of the attenuation is the worst cascade
filter of the fourth order and Sallen-Key structure of
even order filter, too. As is shown, approximatelly
can be written (13)
dBNAOVER 50
(13)
where: N is the number of the biquads. If this
attenuation is over the quantize noise (see Fig.16),
the degradation of magnitude LP filter is not
relevanting, of course.
Figure 16. Low attenuation reduces the signal-to-noise ratio
(SNR)
4. Conclusion
As we can see, only then H-LP and 2T-LP
biquads have a monotone increasing attenuation in
stop band. Another described structures, namely
non-cascading filter structure and/or composition
two (generally even) biquads LP-SK can be used as
well. But it is necessary their attenuation must to be
more, than quantize noise of digital filter.
Another possibility is to combine a filter
structure, where a filter with a finite and small
damping value above the transit frequency is
replaced in one of its elements the structure with a
theoretically infinite attenuation. Thus, one of the
biquads of the SK-LP structure is replaced by one of
the biquads of the H-LP and/or 2T structure.
However, this requires converting the parameters of
the SK structure to an H and/or 2T structure.
Another way is to directly design n-1 biquads in the
SK structure and a single biquad in the H and/or 2T
structure for the n-th order filter, as wel.
For the above reasons, it seems appropriate to
consider the facts described in the article when
choosing a filter and its design.
References
[1] F. Yuan, A. Opal, Computer Methods for Analysis of
Mixed-Mode Switching Circuits, Kluever Academic
Publisher, New York, 2004.
[2] L. Thede, Practical Analog and Digital Filter Design,
Artech House, 2004
[3] S. Winder, Analog and Digital Filter Design, 2nd ed,
Woburn, USA, 2002, pp125-241.
[4] R. Mancini, Op Amps For Everyone – Design Reference,
Texas Instruments, 2002.
[5] J. Puncochar, “Low Pass Filters Sallen and Key With Real
Operational Amplifiers,” Elektrorevue 10, 2005, pp. 1-13.
[6] J.Dostal, Operational Amplifiers. BEN publisher, Prague,
2006.
[7] J. Bicak, M. Leipert, M. Vlcek, A Linear Circuits an
Systems, CTU publisher, Prague, 2007.
[8] T. Dostal, K. Vrba, The Electric filters, PC-DIR, Brno,
1997.
R1R3R4
C2VO
C3C4
R2
C1
VI
R6
R5
_
+
_
+
R7R8
R1R3R4
VO
R2
VI
R6
R5R7R8
rI1 rO1 rI2 rO2
V = 0
LP
filter
SC
filter
SI
filter
digital
filter
LP
filter
antialiasing
filter
reconstruction
filter
fSf(Hz)
L
(dB)
signal
noise
SNR
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.10
Bohumil Brtník
E-ISSN: 2224-266X
98
[9] D. Biolek, Solving electronic circuits. BEN publisher,
Prague, 2004.
[10] W. Jung, Op Amp Applications Handbook, Elsevier,
Oxford, UK, 2005, pp.307-419.
[11] P. Martinek, P. Boreš, J. Hospodka, Electrics Filters, CTU
publisher, Praque, 2003.
[12] K. Hájek, J. Sedláček, Frequency Filters, Ben publisher,
Prague, 2006
[13] V.Mužik, O.Vetchý, T.Šimek, Electronic systems, CTU
publisher, Praque, 2001.
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.10
Bohumil Brtník
E-ISSN: 2224-266X
99
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US