Download Log-Domain Filters Based On LC Ladder Synthesis

Log-Domain Filters Based On LC Ladder Synthesis
D. Perry and G. W. Roberts
Microelectronics and Computer Systems Laboratory
Department of Electrical Engineering
McGill University
3480 University St., Montreal, QC, Canada, H3A 2A7
Abstract. A design method is proposed for the synthesis of linear, high-order, continuous-time filters using a
unique translinear integrator circuit. Unlike previous
attempts at incorporating translinear circuits into filter design, the proposed theory makes explicit use of
the exponential nature of the bipolar transistor. This
technique is based on the operational simulation of LC
ladders. A 5th-order Chebyshev filter is designed, simulated and verified experimentally. The filter shows
good amplitude response as well as distortion levels
comparable to other filtering schemes.
here we show how one can utilize non-linear integrators
and achieve the same overall linear response.
We now introduce two very important operators, LOG and
ANTI-LOG, which help explain the log-domain systems.
They are based on a key circuit, the log-domain cell,
shown in Fig. 1. Analysis of this circuit is left to the
reader, but will result in the basic log-domain equation,
IB = K ⋅ Io ⋅ e
(1)
These two operators, both which can be performed by the
basic cell of Fig. 1, are shown below;
I. Introduction
An area of filter theory which has been of particular interest recently is the design for high-speed, low voltage
applications. One circuit which shows great promise in
these areas is the log-domain filter [1,2]. Based on the
translinear principle [3], these filters make explicit use of
the exponential nature of the bipolar transistor. Because
the need to linearize the transistor has been eliminated,
log-domain filter circuits are simpler than other filtering
schemes. A second advantage is that all of the nodes along
the signal path are low impedance. This serves to keep the
time constants along the signal path low and helps maximize the operating bandwidth of the circuit.
( V A – V B ) ⁄ 2V T
ANTILOG ( V ) = I o ⋅ e
LOG I o ⋅ e
V ⁄ 2V T
V ⁄ 2V T
(2)
(3)
= V
The most important property of these functions is their
inverse nature, namely, LOG [ ANTILOG ( V ) ] = V . We
can now use these operators to show the general form of a
log-domain block. These new systems are based on traditional linear systems but with ANTI-LOG blocks placed at
the inputs and LOG blocks placed at the outputs, as shown
in Fig. 2(a). The obvious drawback of such a system is
that it is no longer linear. Therefore, we need to find a way
to linearize the log-domain system and regain the original
transfer function.
The log-domain filters that can be found in the literature
consist of biquads and a cascade of biquads. Higher-order
filters are introduced in [2] but the design of an Nth order
filter necessitates working with a set of N non-linear equations. Such an approach can quickly become unmanageable. In this paper, we present a new theory of design based
on the operational simulation of LC ladders. In addition to
maintaining the good sensitivity properties of LC ladders,
the complexity of design is greatly reduced.
^
Vi1
Anti
^
ViN
Anti
Vi1
Vo
Linear
System
Log
^
Vo
ViN
(a)
Vi1
Log
ViN
Log
Anti
Anti
Linear
System
Log
Anti
Vo
L
A
(b)
K•Io
Vi
L
A
Linear
System
L
L
Linear
System
A
A
Linear
System
IB
VA
VB
Natural
Cancellation
Fig. 1 The basic log-domain cell.
(c)
II. The LOG and ANTI-LOG Functions
Fig. 2 Linearizing a log-domain system.1
We base our design of log-domain filters on the operational simulation of LC ladders [4]. Instead of implementing the integrator portion of the SFG by linear elements,
1.
Variables marked with a circumflex (^) represent variables in the
log domain - as opposed to variables in the usual linear domain
Vo
Consider placing LOG blocks preceding the input and
ANTI-LOG blocks after the output, as shown in Fig. 2(b).
Because of the inverse nature of these functions, the overall result is a linear input-output relationship. A second
way to get the non-linearities to cancel is to simply join
different log-domain sections together, as shown in Fig.
2(c). This natural cancellation is what makes these logdomain circuits so powerful, and is indeed the basis upon
which this paper was written. Suppose, as will be shown
in this paper, that we can build log-domain circuits of the
form shown in Fig. 2(b), which perform the basic functions of summation, integration and multiplication. Then,
we simply need to join the different blocks together in the
required loops, add the inverse functions at the input and
output, and we get the desired transfer function.
Some simple rules for transforming a linear SFG consisting of summers, multipliers and integrators into a logdomain SFG are described below:
 Vˆ – Vˆ  ⁄ 2V
i
o
T
1

d
C Vˆo = Kˆi I o e
1
dt
 Vˆ – Vˆ  ⁄ 2V
i
o
T
N

+ … + Kˆi I o e
(4)
N
Vˆo ⁄ 2V T
Multiplying through by e
, rearranging, and rewriting the derivative using the chain rule, leads to:
C2V T d
Vˆo ⁄ 2V T
-------------- ⋅
Io e
Io
dt
= Kˆi I o e
Vˆi ⁄ 2V T
1
1
+ … + Kˆi I o e
Vˆi ⁄ 2V T
N
(5)
N
The factor I o ⁄ 2V T can be incorporated into a new constant Ki, such that:
Kˆ i ⋅ I o
K i = -------------2V T
(6)
This scaling factor will be important when transforming
the LC ladder into a log-domain filter. It can either be
incorporated into the multiplication factor Ki as was
shown here or it can be used to scale the capacitors and
inductors as will be shown later. It is this factor which
accounts for the good tunability properties of log-domain
filters. Substituting the new constant into Eq. (5) leads to:
Vˆi ⁄ 2V T
Vˆi ⁄ 2V T
1) Place a LOG block after each integrator
2) Place an ANTI-LOG block at the input to each summer
(before the multiplier)
3) Place an ANTI-LOG block at the output
4) Place a LOG block at the input
We now use the LOG and ANTI-LOG operators defined
previously to rewrite Eq. (7) as:
An example of this will be shown in Section IV.
C
III. The Basic Log-Domain Building Blocks
This section will introduce the different blocks which perform the summing, integration and multiplication operations necessary for log-domain filter design. We pay
particular attention to the input and output sections since
these play an important role in linearizing the overall system.
^
Ki1•Io
C⋅
Vˆo ⁄ 2V T
d
Io e
dt
d
ANTI  Vˆo 
dt
= Ki Io e
1
1
+ … + Ki Io e
N
(7)
N
= K i ANTI  Vˆi  + … + K i ANTI  Vˆi  (8)
1
1
N
N
For the final form of the equation we isolate Vˆo ,
1
Vˆo = LOG ---- ∫ K i ANTI  Vˆi  + … + K i ANTI  Vˆi  dt (9)
1
1
N
N
C
The SFG of this log-domain system is shown in Fig. 3(b).
As expected, we can see the linear system at the centre
with ANTI-LOG and LOG blocks at the input and output
respectively.
A similar analysis applies to the non-inverting integrator
shown in Fig. 4.
^
Vi1
➋
^
KiN•Io
^
Vi1
^
Vo
^
Vo
C
^
Ki1•Io
^
ViN
C
^
V
iN
(a)
^
Vi1
Anti
+
^
ViN
Anti
^
KiN•Io
_1 (.)dt
C∫
Ki1
Log
^
Vo
KiN
Fig. 4 The inverting integrator.
(b)
Fig. 3 The log-domain non-inverting integrator.
The Non-Inverting Integrator: We begin with the multiple-input non-inverting integrator, whose circuit is shown
in Fig. 3(a). Using Eq. (1) and applying KCL at node ➋,
we can write the equation:
Damping: The simplest way to create a damped integrator
is to feed the output of a system back to the input. This is
represented by the SFG shown in Fig. 5(a).
The circuit would then look like the log-domain integrator
shown in Fig. 3(a) but with the output connected back to
the input. Recognizing that this is another instance of the
log-domain cell, we can write the current flowing into the
capacitor as,
I damp = K̂ damp ⋅ I o ⋅ e
 ˆ
ˆ
 V o – V o  ⁄ 2V T
(10)
IV. Design of a 5th-Order Chebyshev Filter
which simplifies to,
I damp = K̂ damp ⋅ I o
(11)
In other words, damping can be represented by a current
source, as shown in Fig. 5(b).
^
Vi1
Anti
_1 (.)dt
C∫
Ki1
+
^
ViN
Anti
We now have all the basic tools necessary for log-domain
filter design. In the next section we will show how to use
these techniques to design a fifth-order lowpass filter.
^
Vo
Log
^
ViI
KiN
KiI = Kdamp
We wish to design a Chebyshev filter which has a cutoff
frequency of 50 kHz and a 1 dB ripple. The LC ladder for
such a design is shown in Fig. 7. Next, a SFG is derived
from the LC ladder [4]. The SFG is modified according to
the rules described in Section II to give the log-domain
SFG shown in Fig. 8. Due to the inverse nature of the
LOG and ANTI-LOG operators, the reader can see how
the overall linear transfer function has been maintained.
(a)
^
Ki1•Io
(b)
Fig. 5 The damped log-domain non-inverting integrator.
Input and Output: In order to keep the overall system linear, we have shown that a LOG block must be placed at
the inputs as well as ANTI-LOG blocks at the outputs.
This procedure is demonstrated for the simple log-domain
system shown in Fig. 6(a). The reader will find that the
resultant input and output circuits are applicable to the
majority of log-domain systems.
Log
Anti
L2 = 3.48 µH
C3 = 9.56 µF
+
VL
-
RL
C5
L4 = 3.48 µH
C5 = 6.80 µF
RL = 1 Ω
IS
LOG
ANTI-LOG
1/RS
-1/RS
Log
Anti
KiN
Ioutput
_1
∫ (.)dt
C1 1
_
∫ (.)dt
L2
Io
C
+1
_1
∫ (.)dt
C3
+
+1
+
-1
^
V4
-1
+
-1/RL
_1
∫ (.)dt
C5
_1
∫ (.)dt
L4
^
V3
+1
+
+1
^
V5
-1
IL
Fig. 8 Log-domain SFG of the Chebyshev filter.
V. Simulation and Experimental Results
(a)
^
Kinput•Iinput
^
V2
-1
+
1
^
Voutput
_1 (.)dt
C∫
Kinput
+
^
ViN
C3
We now replace the different sections of the log-domain
SFG by the appropriate circuits from Section III. This
results in the final circuit shown in Fig. 9. The log-domain
component values were found by scaling the LC ladder
components by the factor I o ⁄ 2V T , where Io=180 µA and
VT = 25mV.
^
V1
^
Vinput
Anti
V3
L4
Fig. 7 LC ladder for 5th-order Chebyshev filter.
^
Vo
C
Iinput
C1
RS = 1 Ω
C1 = 6.80 µF
^
Kdamp•Io
I4
V2
V1
L2
+
−
VS
^
Vi1
I2
RS
Anti
Ioutput
(b)
Fig. 6 Input and output sections.
The integrator is implemented using the circuit of Fig.
3(a), while the LOG and ANTI-LOG blocks come from the
basic log-domain cell of Fig. 1. As in the damped case,
the resultant circuit can be simplified, giving the final
form shown in Fig. 6(b).
Fig. 10 plots both the simulated and experimental performance of the 5th-order Chebyshev filter derived in
Section IV versus its desired frequency response. The
simulation results were found by performing HSPICE AC
analysis on the circuit of Fig. 9. This test was repeated
using both ideal transistor models and models for the
Gennum GA911 bipolar transistors. Unfortunately, AC
analysis is somewhat limited since it relies on linearizing
the non-linear elements of the circuit; thus negating its
basic translinear nature. Multitone testing, a form of
large-signal frequency analysis based on HSPICE transient analysis was therefore used to confirm the AC
results with good success.
The log-domain theory was also tested experimentally by
fabricating a 5th-order Chebyshev filter using a semi-cus-
416, 1993.
[3] Gilbert, B., “Translinear Circuits: A Proposed Classification”, Electron. Lett., vol. 11, pp. 14-16, 1975
[4] A. S. Sedra and P. O. Brackett, “Filter theory and
design: active and passive”, Matrix Publishers, Inc.,
Portland, USA, 1978
[5] J. Silva-Martinez, M. Steyaert, W. Sansen, “Highperformance CMOS continuous-time filters”, Kluwer
Academic Publishers, Boston, 1993
tom bipolar design process provided by Gennum Corp.
(GA911). Fig. 10 shows how the log-domain filter has the
desired response when ideal transistors are used but
exhibits a slightly different response when real transistors
are substituted. This is primarily due to the finite β which
causes base current loss and thus affects the cutoff frequency as well as the passband ripple. Despite these differences the results show that log-domain filtering is
indeed a viable alternative to traditional filter methods.
The integrated log-domain filter was also tested for linearity by measuring its total harmonic distortion (THD) for a
4 kHz input tone of varying amplitude. The plot of THD
versus input amplitude is shown in Fig. 11. These distortion levels are consistent with other filtering schemes
which have quoted peak distortion levels of around -60
dB [5].
0
−10
Desired Response
−20
−30
Gain [dB]
−40
VI. Conclusion
Experimental Results
−50
−60
−70
A new technique for the design of log-domain filters was
introduced. The technique is based on the operational simulation of LC ladders and is considerably simpler than
previous methods. It was used to design a 5th-order
Chebyshev filter with a cutoff frequency of 50 kHz. Simulation as well as the first ever experimental results for a
log-domain filter showed the desired amplitude response
as well as distortion levels consistent with other state-ofthe-art filtering schemes.
−80
−90
−100
0
HSPICE Simulation
_ _
Ideal Txs
• • • Gennum Txs
0.2
0.4
0.6
0.8
1
1.2
Frequency [Hz]
1.4
1.6
1.8
Fig. 10 Frequency response of the log-domain filter.
50
45
THD [Negative dB]
VII. Acknowledgements
This work was supported by NSERC and by the Micronet,
a Canadian federal network of centres of excellence dealing with microelectronic circuits and systems.
40
35
References
Ibias = 180 µA
30
[1] R. W. Adams, “Filtering in the log domain”, Preprint
#1470, presented at the 63rd AES Conference, New
York, 1979.
[2] D. Frey, “Log domain filtering: An approach to current mode filtering”, IEE Proc., vol. 140, pp. 406-
Io
IS/RS
Io
25 −6
10
−5
−4
10
10
Iin [A]
Fig. 11 THD vs. input current.
Io
Io
Io
IL
Io/RS
Io/RL
RS = 1 Ω
CC1 = 24.4 nF
CL2 = 12.6 nF
CC3 = 34.4 nF
^
CC3
^
CL2
^
CC1
Io
Io
2
5
x 10
Io
^
CL4
^
CC5
Io
Fig. 9 Circuit diagram of the 5th-Order Chebyshev log-domain filter.
CL4 = 12.6 nF
CC5 = 24.4 nF
RL = 1 Ω