HIGH-ORDER LOWPASS AND BANDPASS ELLIPTIC
LOG-DOMAIN LADDER FILTERS
E.M.Drakakis†, A.J.Payne†,C.Toumazou†, A.E.J.Ng‡ and J.I.Sewell‡
† Dept. of El.&El.Eng., Imperial College, Exhibition Road, London, SW7 2BT, UK
‡Dept. of El.&El.Eng., University of Glasgow, Glasgow, G12 8QQ, UK
{e.drakakis, a.j.payne, c.toumazou}@ic.ac.uk, {a.ng, j.sewell}@elec.gla.ac.uk
ABSTRACT
This paper reports a multitude of HSPICE simulation results
corresponding to high-order (fifth, sixth and tenth) differential classA log-domain lowpass and bandpass ladder filter topologies. The
designs are elliptically approximated and were obtained by means of
an appropriately extended version of the XFILTER filter compiler.
The non-desirable effect of the BJT finite beta is shown; circuit
modifications mitigating - to a certain extent - the device nonideality-induced impact are also reported.
1. INTRODUCTION
Log-domain filtering [1] is characterised by considerable
mathematical burden originating from the fact that the active devices
(usually BJTs) are treated as non-linear computational primitives
allowing the processing of large input current signals which strongly
exercise the non-linear (exponential) I-V device characteristic while
preserving the input-output large-signal linearity. The mathematical
complexity associated with the technique has, so far, restricted the
reported externally-linear-internally-non-linear (ELIN) [2] logdomain topologies to all-pole single-ended signal-flow-graph
(SFG)-based ladder designs [3], two-port cascades [4], maximally
flat responses [5] or cascaded biquads with or without feedback [68]. Recent advances [9] in the development of the XFILTER CAD
package [10] however, have allowed - for the first time - the
automated synthesis of high-order log-domain ladder filters which
comply with predetermined specifications. XFILTER offers a
variety of commonly used and arbitrary frequency response
approximations and matrix decompositions [10]. This progression
allows for the viability of the log-domain technique to be evaluated
in a systematic way. More specifically high-order differential elliptic
ladder filter topologies can be studied for the first time. This paper
presents a plethora of HSPICE simulation results which extend our
understanding of log-domain filtering since they: a. indicate the way
in which the device non-idealities and particularly the finite BJT
current gain affects the intended frequency response of demanding
high-order (fifth, sixth and tenth) elliptically approximated lowpass
or bandpass ladder designs, and b. reveal to a certain extent the way
in which these high-order designs behave in general. Possible circuit
modifications (suitable for bipolar processes) mitigating the device
non-ideality-induced undesirable effects on the pre-specified
transfer functions are also reported. Large-signal transient
simulations verifying the reported small-signal ac behaviours were
also performed.
2. A FIFTH-ORDER CLASS-A DIFFERENTIAL LOW-PASS
ELLIPTIC LADDER FILTER
The netlist of a fifth order differential elliptically approximated
ladder filter was produced by means of XFILTER. The Appendix
provides part of such a netlist corresponding to the interconnection
of the basic blocks termed “fde”, “fdintcap” and “fdidc” (see Fig.1);
XFILTER uses these elementary blocks to realise the desired
frequency responses.
Voutp
Voutn
fde
I0
I0
fdintcap
Vinp
Vinn
I0
I0
Id C
trimming
current
C Id
fdidc
trimming
current
Fig.1 A differential log-domain integrator [11]
It should be underlined however that the netlist provided in the
Appendix is indicative only, corresponds to an ideal case (beta >> 1,
capacitor values in the range of nFs) and simply aims to serve as an
example of the functionality of the extended XFILTER code. The
simulation results which follow though, were obtained using
commercially available process parameters. When the basic blocks
fde, fdintcap and fdidc are assembled [11] a differential log-domain
integrator is produced. As it will become clear, a considerable
improvement of the simulated responses can be achieved once the
diode-connections are substituted for BJTs (as shown in Fig.1). This
simple buffering scheme seems to somewhat mitigate the betainduced errors of the topology (1/β errors at the buffering nodes
now become proportional to 1/β2). Fig.2 reveals in a comparative
way the effect of beta upon the fifth order elliptic response (the rest
of the nominal transistor parameter values were kept unaltered when
beta was changed): higher beta values lead to responses closer to the
ideal one. However, when the buffering connections are introduced,
a considerable improvement can be observed when beta=200 (see
Fig.3): the passband is situated close to the ideal one whereas the
notch depth increases (compare with Fig.2). Curves (1) and (2)
correspond to different dc bias current values present at the input
(denoted hereafter as D). For curve (1) D=85µA whereas for curve
(2) D=8.5µA. Fig.4 illustrates in detail the passband ripple both for
the ideal case and when beta=200 (curve (1)); small (∼ 16nAs)
trimming currents fed into the integrating nodes (see Fig.1) can
compensate for the Q-degradation as shown by curve (2) in the same
figure. Fig.5.a shows an example of tuning via current (one octave
up in frequency (curve (3)) and octave down (curve (2)) for a
response with a cut-off frequency of 1MHz (curve (1)); curve (4) of
the same figure corresponds to curve (1) when compensated by
means of small trimming currents (∼ 0.2µA). Figure 5.b shows the
output current THD levels for curve (4) of Fig.5.a when the
modulation index M.I. (defined as the ratio of the peak amplitude to
the dc bias current D) varies between 10% and 90% for three
different frequencies (0.1, 0.5 and 0.9 MHz) situated in the
passband; 1000 points per period were used with the simulation
accuracy set to the maximum allowable level. The results can be
characterised as satisfactory; for input M.I. of 50 to 60% the THD
distortion level remains below -60dBs.
-40
THD (dBs)
-50
-60
-70
100KHz
-80
500KHz
-90
900KHz
-100
0
10
20
30
40
50 60
70
80
90 100
M .I.(%)
Fig. 5.b Passband THD level (dBs) for curve (4) of Fig.5.a
3.SIXTH AND TENTH ORDER CLASS-A DIFFERENTIAL
BAND-PASS ELLIPTIC LADDER FILTERS
The beta-induced error reduction scheme mentioned previously does
seem to offer a viable solution when the capacitor and bias current
value spreads are low; for the fifth order LP elliptic design for
example, the capacitor and current spread values were 2.5 and 17
respectively. In contrast, when high spread designs are generated,
buffering does not seem to improve the frequency response. More
specifically: considering a sixth-order bandpass elliptic ladder
response with approximate capacitor spread value of 10.3 and
current spread value higher than 100 and referring to Fig.6, it can be
easily deduced that despite the incorporation of buffering no
improvement can be observed (for clarity the ideal response is
presented shifted in the frequency domain). On the other hand, for
the case of lower capacitor and current value spread designs, the
desirable effect of buffering can be clearly noticed again as shown in
Fig.7 which illustrates a sixth-order bandpass elliptic response with
approximate capacitor and current spread values of 1.5 and 31
respectively; referring to the same figure, the passband loss of curve
(2) is rectified and the position of the transmission zeros becomes
clearer. Fig. 8 illustrates in detail a similar sixth-order bandpass
elliptic response tuned via current to a higher center frequency;
extensive simulations revealed that the “tilt” of the pass-band can be
mitigated by fine tuning the current value of a specific fde block. In
a similar way, the transmission zeros can also be leveled-up. Fig.9
deals with a third elliptic bandpass design of sixth-order
characterised by even lower capacitor and current spread values (1.4
and 15 respectively): the desirable impact of buffering is verified
once more. The slight tilt of the passband can be corrected by means
of a combined effect of small trimming currents and small resistors
connected in series with the integrating capacitors. Curves (2) and
(3) of Fig. 10 show the effect of the presence of trimming currents
only whereas curves (4) and (5) of the same figure illustrate the
impact of series-connected resistors only; the response (6) is
obtained for a specific trimming current-series resistor value
combination which corrects the passband tilt. Large-signal transient
simulations showed that no severe penalty was paid as far as
linearity (IMD product levels) is concerned. Table I provides
indicative simulation results; for two fundamental tones of 0.98 and
1.02 MHz (situated 2% apart from the center frequency) with a
combined M.I. corresponding to 86%, the IMD product levels were
≈-40dBs. For convenience Table II summarises the capacitor and
bias current spread values of the sixth-order bandpass topologies
presented so far.
M.I. (%)
IMD(dBs)
28
-56
57
-44
86
-40
Table I IMD product levels for the 6th-order BP response of Fig.10;
fundamental tones: 0.98 and 1.02 MHz
capacitor
current
BP filter
spread
spread
Fig.6
10.3
>100
Figs.7,8
1.5
31
Figs.9,10
1.4
15
Table II Capacitor and bias current spread values for the 6th-order
bandpass designs
Fig.11 considers the frequency response of a tenth order differential
bandpass elliptic ladder topology for both ideal and realistic process
parameters. The design is characterised by a capacitor value spread
of 1.58 and a bias current spread value of 15; the center frequency
equals to 1MHz, the passband extends to 200KHz, the passband
ripple amounts to 0.5dBs whereas the stopband rejection is more
than 70dBs. The deviation from the ideal case is disappointing when
beta-compensation is not incorporated. However, as illustrated in
Fig.12, an appropriate combination of trimming currents and series
connected resistors (found by means of exhaustive simulations) in
conjunction with a limited trimming (less than 5% of the nominal
value) of a specific fdintcap block capacitor pair value, can lead to a
considerable overall improvement for a similar design. Table III
provides indicative simulation results; for two fundamental tones of
1.6 and 1.7 MHz (situated 3% apart from the center frequency
which equals 1.65MHz) with a combined M.I. corresponding to
80%, the output current IMD levels were ≈ -38dBs.
M.I. (%)
IMD(dBs)
40
-44
60
-41
80
-38
Table III IMD product levels for the 10th-order BP response of
Fig.12; fundamental tones: 1.6 and 1.7 MHz
4. SUMMARY
This paper dealt with the presentation of HSPICE simulation results
corresponding to differential class-A fifth-order lowpass, sixth-order
bandpass and tenth-order bandpass ladder log-domain filters. In
every case the elliptic approximation was used. The filters were
synthesised by virtue of an appropriately extended version of the
XFILTER CAD package. Examples of the impact of the finite BJT
current gain were provided. Moreover, simple circuit modifications
leading to improved responses were reported. Test-chips are
currently being laid-out.
5. ACKNOWLEDGEMENTS
Financial support from the Engineering and Physical Sciences
Research Council (GR/M45368) is gratefully acknowledged.
6. REFERENCES
[1] D.R.Frey, “Exponential State-Space Filters: a generic currentmode design strategy”, IEEE Trans. CAS-I, vol.43, no.1pp. 3442, 1996
20
beta=2000
beta=200
10
0
-10
ideal response
TF (dBs)
-20
-30
-40
-50
-60
-70
-80
2
10
3
4
10
5
10
10
Freq (Hz)
6
7
10
10
8
10
Fig.2 High-beta values lead to behaviours close to the ideal
response; capacitors in the range of hundreds of pFs; current
source values in the range of tens of µAs; power supply levels:
±1.65V
20
10
ideal response
(2)
0
(1)
-10
-20
TF (dBs)
iinp vdd 17 dc= 8.49e-05 ac= 4.2e-05 0.0
iinn vdd 18 dc= 8.49e-05 ac= 4.2e-05 180
xfdintcap1 1 2 fdintcap c= 4.9e-08
xfdintcap2 3 4 fdintcap c= 8.6e-08
xfdintcap3 5 6 fdintcap c= 3.5e-08
xfdintcap4 7 8 fdintcap c= 8.6e-08
xfdintcap5 9 10 fdintcap c= 8.2e-08
xfdidc6 11 12 fdidc io= 7e-05
xfdidc7 13 14 fdidc io= 1e-04
xfdidc8 15 16 fdidc io= 5.7e-05
xfdidc9 1 2 fdidc io= 4.7e-05
xfdidc10 5 6 fdidc io= 2.9e-05
xfde11 7 8 1 2 fde io= 1e-04
xfde12 7 8 3 4 fde io= 6.9e-05
xfde13 9 10 3 4 fde io= 1e-04
xfde14 9 10 5 6 fde io= 1e-04
xfde15 1 2 8 7 fde io= 6.8e-05
xfde16 3 4 8 7 fde io= 1e-04
xfde17 3 4 10 9 fde io= 1e-04
xfde18 5 6 10 9 fde io= 6.1-05
xfde19 13 14 12 11 fde io= 6.1e-05
xfde20 11 12 14 13 fde io= 5.9e-06
xfde21 15 16 14 13 fde io= 1.8e-05
xfde22 13 14 16 15 fde io= 4.2e-05
xfde23 13 14 2 1 fde io= 2e-05
xfde24 11 12 4 3 fde io= 8.3e-06
xfde25 15 16 4 3 fde io= 2.6e-05
xfde26 13 14 6 5 fde io= 4.2e-05
xfde27 7 8 11 12 fde io= 1e-04
xfde28 7 8 13 14 fde io= 4.9e-05
xfde29 9 10 13 14 fde io= 7e-05
xfde30 9 10 15 16 fde io= 1e-04
xfde31 1 2 12 11 fde io= 4.7e-05
xfde32 5 6 16 15 fde io= 2.9e-05
xfde33 17 18 1 2 fde io= 8.5e-05
xfde34 17 18 11 12 fde io= 8.5e-05
q35 17 17 0 0 nbjt
q36 18 18 0 0 nbjt
q37 19 5 0 0 nbjt
q38 20 6 0 0 nbjt
eoutp 19 0 vcvs vdd 0 1.0
eoutn 20 0 vcvs vdd 0 1.0
.probe ac i(eoutp) i(eoutn)
-30
-40
-50
-60
-70
-80
2
10
3
10
4
10
5
10
Freq (Hz)
6
7
10
10
8
10
Fig.3 The buffering improves (compare with Fig.2) the desired
response when beta=200; power supply levels: ±1.65V
4
3
2
(2) (with trimming currents)
1
TF (dBs)
[2] Y.Tsividis, “Externally Linear Time Invariant Systems and
their application to companding signal-processors”, IEEE
Trans. CAS-II, vol.44, no.2, pp. 65-85, 1997
[3] V.W.Leung and G.W.Roberts, “Effects of Transistor
Nonidealities on High-Order Log-domain ladder Filter
Frequency Responses”, IEEE Trans. CAS-II, vol.47, no.5,
pp. 373-387, 2000
[4] G.van Ruymbeke, C.C.Enz, F.Krummenacher and M.Declerq,
“A BiCMOS programmable continuous-time imageparameter method synthesis and voltage-companding
technique”, IEEE JSSC, vol.32, no.3, pp.377-387, 1997
[5] M.N.El-Gamal and G.W.Roberts, “Very High-Frequency Logdomain Bandpass Filters”, IEEE Trans. CAS-II, vol.45, no.9,
pp. 1188-1198, 1998
[6] E.M.Drakakis, A.J.Payne and C.Toumazou, “Multiple
Feedback Log-domain Filters”, in Proc. IEEE ISCAS98, vol.1,
pp. 317-330
[7] E.M.Drakakis and A.J.Payne, “Leapfrog Log-domain Filters”,
in Proc. IEEE ICECS 1998, vol.2, pp.385-388
[8] D.R.Frey, “Log-domain Filtering: an approach to current-mode
filtering”, IEE Proc. Part-G, vol.140, pp.406-416, 1993
[9] A.E.J.Ng, J.I.Sewell, E.M.Drakakis, A.J.Payne and
C.Toumazou, “A Unified Matrix Method for systematic
Synthesis of Log-domain Ladder Filters”, submitted to IEEE
ISCAS 2001
[10] A.E.J.Ng and J.I.Sewell, XFILT reference manual, ver. 3.1.
Dept.of El. and El.Eng., Univ.of Glasgow, 1997
[11] F.Yang, C.Enz and G.van Ruymbeke, “Design of Low-Power
and Low-Voltage Log-domain Filters, in Proc. IEEE ISCAS96,
vol.1, pp. 117-120
APPENDIX
A sample XFILTER netlist corresponding to a fifth-order lowpass
elliptic design:
0
ideal response
-1
-2
(1)
-3
-4
3
10
4
10
Freq (Hz)
Fig.4 The application of small trimming currents (∼16nAs) can
lead to considerable improvement in the passband; power
supply levels: ±1.65V
10
10
ideal response
(4) (with trimming currents)
0
0
-10
(2)
with buffering
(3)
(1)
-10
-20
without buffering
TF (dBs)
TF (dBs)
-30
-20
-40
-30
-50
-60
-40
-70
-50
-80
-60
5
10
6
7
10
-90
5
10
8
10
10
6
Fig.5.a Tuning of 1MHz response (curve(1)) via current; capacitors
in the range of tens of pFs; current source values in the range of
tens of µAs; power supply levels: ±1.65V
7
10
Freq (Hz)
Freq (Hz)
10
Fig.9 Very low spread sixth-order bandpass elliptic design
verifying the desirable impact of buffering. Operational conditions
same as in Figs.7 and 8
5
20
shifted ideal response
(80nA,0 Ohm) (3)
0
(60nA,0 Ohm) (2)
(50nA,12 Ohm) (6)
(0nA,0 Ohm) (1)
0
(4)(0nA,10 Ohm)
(5)(0nA,20 Ohm)
-5
without
buffering
-40
TF (dBs)
TF (dBs)
-20
-60
-10
with
buffering
-80
-15
-100
-120
1
10
2
3
10
10
4
5
10
6
10
7
10
-20
8
10
6
10
Freq (Hz)
10
Freq (Hz)
Fig.6 Sixth-order bandpass elliptic response; capacitors in the
range of tens of pFs; current value spread > 100
10
0
10
-10
TF (dBs)
Fig.10 Effects of small trimming currents and small resistors
connected in series with the integrating capacitors for the response
of Fig.9
(a)
-20
0
ideal response
-30
-40
-10
without buffering
-50
-20
-60
-70
3
10
4
-30
10
Freq (Hz)
(1) with buffering
0
TF (dBs)
-10
-40
TF (dBs)
10
-50
(b)
-60
-20
(2) without buffering
-30
-70
-40
-50
-80
-60
-90
-70
6
10
Freq (Hz)
Fig.7 Low-spread sixth-order elliptic response; (a): shifted ideal
frequency response (b): improved response when buffering is
applied; capacitors in the range of tens of pFs; current source
values in the range of µAs; power supply levels: ±1.65V
-100
5
10
7
10
Fig.11 Very low spread 1MHz tenth-order bandpass elliptic
response; capacitor values in the range of tens of pFs and current
source values in the range of µAs; power supply levels: ±2V
10
0
5
-2
TF (dBs)
(a)
TF (dBs)
6
10
Freq (Hz)
-4
-6
0
(a)
-5
-10
-8
-15
-10
2.5
3
3.5
Freq (Hz)
4
-20
6
10
4.5
Freq (Hz)
6
x 10
0
0
-10
-20
(b)
TF (dBs)
TF (dBs)
-20
-30
-40
(b)
-40
-60
-50
-80
-60
-70
6
10
7
10
Freq (Hz)
Fig.8 Sixth-order bandpass elliptic response with a center
frequency of 3.4MHz and a passband of ∼340KHz. (a):passband
detail (b):the complete response; buffering is applied; spread values
and operational conditions same as in Fig.7
-100
5
10
6
10
Freq (Hz)
7
10
Fig.12 A beta-compensated tenth-order bandpass elliptic response
with small trimming currents (∼90nAs) and small resistors (∼11Ω)
connected in series with the integrating capacitors; operational
conditions as in Fig. 11; center frequency=1.65MHz,
passband≈400KHz. (a):passband detail (b): the complete response