Adaptation of Log Domain Second Order Filters
Implemented by Floating Gate MOSFETs
Yiming Zhai and Pamela A. Abshire
Department of Electrical and Computer Engineering
University of Maryland
College Park, MD 20742, USA
ymzhai, pabshire@umd.edu
Least Mean Square (LMS)-based adaptation algorithms.
LMS methods are well suited to implementations of FIR
filters. In this paper we present alternative methods based on
Lyapunov stability well suited for adaptive control of
Infinite Impulse Response (IIR) filters. IIR filters offer the
advantage of smaller filter structures and fewer filter
coefficients than FIR filters in order to model plants of
similar complexity.
This work extends our previously reported design of an
adaptive log domain first order low-pass filter [7]. We
derive novel robust control laws based on the Lyapunov
method for a second order filter and simplify them for
compactness and VLSI-friendliness in the implementation
of the learning rules. The structure of the second order filter
using floating gate MOSFETs is also a new approach based
on the first order low-pass filter design.
Section II develops robust learning rules for adapting the
quality factor and time constant of a log domain second order
filter. Section III describes the implementation of the log
domain filter structure and integration of the learning rules
for quality factor and time constant. Section IV presents
simulation results of the adaptive system for a variety of
inputs which are suitable for model reference estimation.
Section V summarizes this work.
Abstract—We describe an adaptive log domain second order
filter with integrated learning rules. The system is
implemented using multiple input floating gate transistors to
realize on-line learning of quality factor and time constant. We
use adaptive dynamical system theory to derive robust control
laws for both quality factor and time constant adaptation in a
system identification task. The log domain filters adapt to
estimate the quality factor and time constant of a reference
filter accurately and efficiently as the parameters of the
reference are changed. We present simulation results for
0.5µm technology which demonstrate that adaptation occurs
within milliseconds. The filter consumes 2µW and the entire
system consumes 10µW.
I.
INTRODUCTION
Adaptive signal conditioning offers the opportunity to
reject noise and improve signal quality dynamically in a
variety of complex and demanding applications. To achieve
this, control laws must use limited information to adjust
parameters of the adaptive system in directions that produce
robust system adaptation. In this paper log domain filter
architecture and floating gate MOSFETs are combined to
realize accurate and stable learning rules for the system
parameters of a bandpass filter, quality factor and time
constant.
Several groups have described filtering applications
based on floating gate MOS circuits. Hasler et al. [1]
described the Auto-zeroing Floating Gate Amplifier (AFGA)
and its use in bandpass filter structures with very low
frequency response capability. Fernandez et al. [2] described
a 1V micropower lowpass filter implemented using floating
gate transistors. Rodriguez-Villegas et al. [3] designed a log
domain integrator based on floating gate transistors. Minch
[4, 5, 6] developed circuits and synthesis techniques using
Multiple Input Translinear Elements (MITEs) for a variety
of signal processing applications. Abshire et al. [7] reported
an adaptive log domain first order low-pass filter using
floating gate transistors.
Few groups have reported integrated analog adaptive
filters. Juan et al. [8] and Stanacevic and Cauwenberghs [8,
9] have designed analog transversal Finite Impulse
Response (FIR) filters that include adaptation of weights.
Both Juan et al. and Stanacevic and Cauwenberghs use
0-7803-9197-7/05/$20.00 © 2005 IEEE.
II.
DERIVATION OF LEARNING RULES
We describe an adaptive filter which addresses the
classical problem of system identification depicted in Fig 1.
Figure 1: The system identification problem
A tunable second order filter is used as the model to identify
an unknown system. An input signal is applied to both an
unknown system (plant) and an adaptive estimator (model)
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system. Control laws are designed using observable outputs
to adjust the parameters of the estimator so as to ensure
stability of the learning procedure.
V& (e) = e&1&e&1 + B 2e1e&1 + e2 e&2 + e3e&3
= − ABe&12 + e2 (− Be&1 x&2 + e&2 ) + L
  &x&


+ e3 e&1  2 − B( x2 − D ) + e&3 

 B

The frequency response of the second order filter is
described by the following equation:
I out ( s) = I in ( s) ⋅
τ ⋅s
1+
1
⋅τ ⋅ s + τ 2 ⋅ s 2
Q
+ I dc ⋅
By choosing the following control laws:
1
1+
1
⋅τ ⋅ s + τ 2 ⋅ s 2
Q
we find that the candidate Lyapunov function has a negative
time derivative:
While the bandpass function eliminates DC components
from the input, log domain filters are inherently current
mode circuits with strictly positive currents. Thus we must
introduce an output bias, denoted here as a constant current
Idc which is independent of the input signal [10].
V& (e) = − ABe&12
If we further assume that x2 is a sinusoidal signal, we
find that − &x&2 can be expressed as the product of (x2 − D )
and a positive scalar. Thus we can incorporate the second
order derivative in the learning rule of e&3 into the term
(x2 − D) . These rules may be simplified further since in
current mode log domain filters the quality factors and time
constants are positive. The positive scalars in the rules affect
the rates of the adaptation, but not the direction. Thus the
rules above can be simplified:
We describe the unknown plant and adaptive model filter
using the state-variable representation:
&x&1 = − B 2 x1 − ABx&1 + Bu& + B 2 D
plant
&x&2 = − B 2 x2 − A B x&2 + B u& + B 2 D
model
where x1 and x2 are the plant and model outputs, A and A
are the reciprocals of plant and model quality factors, B and
B are the reciprocals of plant and model time constants, u
is the input to both filters, and D is the output bias of both
outputs.
output error
e2 = A − A
(1/quality factor) error
e3 = B − B
(1/time constant) error
We estimate the reciprocal quality factor by integrating
the product of the output error derivative and model output
derivative and estimate the reciprocal time constant by
integrating the product of the output error derivative and
model output without bias.
III.
&e&1 = &x&2 − &x&1
e&2 = A&
e&3 = B&
We cannot control the dynamics of the output error since
that depends on the unknown input u, but we can derive
adaptive laws that specify the dynamics of the parameter
errors so that the estimator learns the behavior of unknown
system.
I1 = I 0 e K (V1 +Vr ) = I in
The direct Lyapunov method is employed to derive
appropriate learning rules [11]. We must find a scalar
function which satisfies three conditions: positive definite,
negative definite time derivative and radially unbounded. For
the adaptation of the second order bandpass filter we
consider the candidate Lyapunov function:
V (e ) =
(
1 2
e&1 + B 2e12 + e22 + e32
2
CIRCUIT IMPLEMENTATION
A. Implementation of Log Domain Bandpass Filters
We implement the log domain bandpass filter using
dynamic MITE networks. The circuit in Fig 2 is used both as
the unknown plant and the estimated model with labels in
parentheses representing model parameters. In subthreshold
operation MITE current is an exponential function of the
summed inputs.
We seek control laws which drive all errors toward zero
with time, so we focus our development on the dynamics of
the error system:
e&1 = x&2 − x&1
e&3 ∝ e&1 ( x2 − D )
e&2 ∝ e&1 x&2
An error system is constructed in order to evaluate the
performance and stability of the adaptation.
e1 = x2 − x1
&x& 

e&3 = e&1  B ( x2 − D ) − 2 
B

e&2 = Be&1 x&2
I 4 = I 0e
K (V3 +V4 )
I 7 = I 0e
K (V4 +V5 )
= Iτ
I 2 = I 0 e K (V1 +V2 )
I 5 = I 0e
K (V2 +V5 )
I8 = I 0e
I 3 = I 0 e K (V2 +V3 )
= Iτ
K (V4 +V6 )
I 6 = I 0 e K (V5 +Vr )
I 9 = I 0 e K (V6 +Vr ) = I dc
Kirchoff’s Current Law (KCL) is applied at the capacitive
nodes to obtain the following relationships:
and I 8 = I 7 + CV&4
M6, M7, M8 and M9 form a translinear loop:
Iτ / Q + I 3 = I 2 + CV&2
I 8 = I 7 I dc / I 6
)
I 7 = I 7 I dc / I 6 − CV&4
(1)
M3, M4, M7 and M5 form a translinear loop: I 3 = Iτ2 / I 7
This function satisfies the first and third conditions. To
evaluate the second condition, we evaluate the temporal
derivative of the candidate function:
M1, M2, M5 and M6 form a translinear loop:
I 2 = I in Iτ / I 6 = Iτ / Q + I 3 − CV&2 = Iτ / Q + Iτ2 / I 7 + CI&6 / KI 6
So the input current can be expressed as
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circuit of Fig 3(c) in order to compute I d 3 − I d 4 ∝ x& 2 , since
I& f = I f KV&5 and I f is a positive current which only affects
(2)
I in = I 6 / Q + Iτ I 6 / I 7 + CI&6 / KIτ
Let us consider an intermediate current I x :
I x = I τ I 6 / I 7 = I τ e K (Vr −V4 )
I&x = − I x KV&4
the rate of adaptation.
Iτ / I x = I 7 / I 6
If we substitute equation (1) for I 7 ,
Iτ I 6 = I x ( I 7 I dc / I 6 − CV&4 ) = I x ( Iτ I dc / I x + CI&x / KI x )
C I&x / K = Iτ ( I 6 − I dc )
I x = KI τ ( I 6 − I dc ) / sC
(3)
and further substitute the new expression for Ix into equation
(2), we obtain
I in = I 6 / Q + KIτ ( I 6 − I dc ) / sC + sCI 6 / KIτ
⇒ I6 =
I in ( sC / KIτ ) + I dc
1 + ( sC / KIτ ) / Q + ( sC / KIτ ) 2
which is a second order transfer function with quality factor
Q and time constant τ=C/KIτ. We can easily tune the bias
current Iτ and bias voltage VQ to change the time constant and
quality factor respectively, thus changing the central
frequency and shape of the filter.
Figure 3: (a) and (c) are circuits for computing temporal derivative of
voltage. (b) is the circuit for converting output current difference to a
voltage.
Both the learning rules for quality factor and time
constant require a four quadrant multiplication, implemented
using the MITE circuit shown in Fig 4(a). The labels in
parentheses represent the adaptation circuits for the time
constant. Circuits for integrating the learning rules are shown
in Fig 4(b) and (c). Note that VQ can be higher than Vdd, so
the voltage Vee in Fig 4(b) is higher than Vdd.
Figure 2: Log domain MITE network for a second order filter used for both
plant and filter. Labels in parentheses refer to filter variables, the rest to the
plant.
B. Implementation of Learning Rules
The behavior of plant and model filters is controlled by
two parameters: quality factor and time constant. We have
derived learning rules for the reciprocals of these parameters
in Section II. The inputs to the learning rules are the
temporal derivative of the output difference, the temporal
derivative of the model output and the model output
excluding bias. Fig 3(c) is a circuit for computing temporal
derivative. We use it to implement both the derivatives of the
output difference and the model output.
Figure 4: (a) Circuit for computing four quadrant multiplication. (b)
Integrator circuit for quality factor adaptation. (c) Integrator circuit for time
constant adaptation.
IV.
SIMULATION RESULTS
The circuit is simulated with HSPICE using BSIM3v3
model for a commercially available 0.5µm technology. The
technique in [12] is used to avoid floating-node problems in
the simulation. The voltage source Vdd for both filters is 1.5V
and the voltage sources Vcc and Vee required for adaptation
are both 2.5V. We use a sine wave (Fig 5) and superposition
of sine waves (Fig 6 and 7) as inputs.
The temproal derivative circuit is a wide range OTA that
operates as a voltage follower with a capacitor connected to
the output as shown in Fig 3(a). The output current is Id=Id1Id2 in Fig 3(c). The larger the gain is , the more accurate the
calculation. So we operate the input devices near threshold
for large transconductance. We use the circuit of Fig 3(b) to
compute the filter output error and convert it to a voltage,
then use the circuit of Fig 3(c) to realize the derivative
I d 1 − I d 2 ∝ e&1 . We also use the intermediate node voltage V5
of the model filter in Fig 2 as the input to the derivative
Fig 5 shows adaptation with a 10kHz sine wave as Q and
τ vary. The sine wave is biased at 100nA with an amplitude
of 60nA. Fig 5(a) shows VQ and VQ_est. VQ is varied as 1.47V
from 0-2ms, 1.45V from 2-6ms and 1.46V from 6-10ms. Fig
5(b) shows Vτ and Vτ_est, which correspond to different values
of Iτ. Iτ is varied as 40nA from 0-4ms, 45nA from 4-8ms, and
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filter using a log domain topology which has wide tuning
range and large dynamic range and capability for high
frequency operation. Further, we have developed robust
learning rules for system identification based on the direct
Lyapunov method for the second order filter. We implement
these learning rules using MITE structures, which is compact
and elegant although necessarily more complex than the
design of the adaptive first order low pass filter.
35nA from 8-10ms. Fig 5(c) is the error between the plant
and filter output. For all changes in VQ and Vτ, VQ_est and
Vτ_est track the new values accurately. The error converges to
zero when VQ_est converges to VQ and Vτ_est converges to Vτ.
The adaptation rate depends on signal strength, currents IQa
and Iτa, and capacitances CQ and Cτ.
We are in the process of fabricating these circuits,
experimentally validating these results and extending this
work to more comprehensive adaptive filter structures.
ACKNOWLEDGMENT
We thank the MOSIS service for providing chip
fabrication through their Educational Research Program.
These circuits will be used to teach an undergraduate course
in mixed signal circuit design. We thank Paul Hasler for
stimulating discussions at the Telluride Neuromorphic Eng
Workshop 2004. Y.Z. is supported by Laboratory for
Physical Sciences. This work is supported by an NSF
CAREER Award (NSF-EIA-0238061).
Figure 5: 10kHz sine wave input signal: (a) Quality factor adaptation. (b)
Time constant adaptation. (c) Output error.
Next we show adaptation when the input signal is a
mixture of sine waves. In Fig 6, we use a combination of sine
waves at 10kHz, 20kHz, 40kHz and 80kHz as input. In Fig 7,
we use a summation of 6 sine waves, whose frequency ratio
is an irrational number π / 2 , spanning from 10kHz to 96kHz.
In each case, VQ_est accurately tracks VQ and Vτ_est tracks Vτ,
and the output error approaches zero when adaptation is
finished.
REFERENCES
[1]
Hasler, P., Minch, B.A. and Diorio, C., "An Autozeroing FloatingGate Amplifier," IEEE Transactions on Circuits and Systems II:
Analog and Digital Signal Processing, vol. 48, pp. 74-82, 2001.
[2] Fernandez, R., Lopez-Martin, A.J., de la Cruz, C.A., Carlosena, A.,
“A 1V micropower FGMOS Log-Domain Filter,” International
Conference on Electrons, Circuits and Systems, 2002, vol.1, pp381384
[3] Rodriguez-Villegas, E., Yufera, A., Rueda, A., “A 1-V micropower
log-domain integrator based on FGMOS transistors operating in weak
inversion,” IEEE Journal of Solid-State Circuits, 2004, Vol. 39 ,
pp.256-259
[4] Minch, B.A., Hasler, P. and Diorio, C., "Multiple-Input Translinear
Element Networks," IEEE Transactions on Circuits and Systems II,
vol. 48, pp. 20-28, 2001.
[5] Minch, B.A., "Multiple-Input Translinear Element Log-Domain
Filters," IEEE Transactions on Circuits and Systems II: Analog and
Digital Signal Processing, vol. 48, pp. 29-36, 2001.
[6] Minch, B.A., "Synthesis of Static and Dynamic Multiple-Input
Translinear Element Networks," in IEEE Trans. Circuits and Systems
I, 2004 vol. 51, pp.409-421
[7] Abshire, P.A., Wong, E.L., Zhai, Y. and Cohen, M.H., "Adaptive Log
Domain Filters Using Floating Gate Transistors, " Circuits and
Systems, 2004. ISCAS '04. Proceedings of the 2004 International
Symposium, vol: 1 , 23-26, 2004
[8] Juan, J.-K., Harris, J.G. and Principe, J.C., "Analog Hardware
Implementation of Adaptive Filter Structures," presented at
International Conference on Neural Networks, 1997.
[9] Stanacevic, M. and Cauwenberghs, G., "Charge-Based Cmos Fir
Adaptive Filter," presented at Midwest Symposium on Circuits and
Systems, 2000.
[10] Edwards, T.R., "Time-Frequency Acoustic Processing and
Recognition: Analysis and Analog VLSI Implementations," Chapter 3,
PhD dissertation, 1999
[11] Narendra, K.S. and Annaswamy, A.M., Stable Adaptive Systems.
Prentice-Hall, New Jersey, 1989.
[12] Rahimi, K., Diorio, C., Hernandez, C., Brockhausen, M.D., "A
Simulation Model for Floating-Gate Mos Synapse Transistors,"
presented at ISCAS, 2002.
Figure 6: Four harmonic sine waves input signal: (a) Quality factor
adaptation. (b) Time constant adaptation. (c) Output error.
Figure 7: Six geometrically spaced sine waves from 10-96kHz input signal:
(a) Quality factor adaptation. (b) Time constant adaptation. (c) Output error.
V.
CONCLUSIONS
We have developed a circuit design approach for log
domain adaptive filters that extends earlier work from
adaptation of first order low-pass filters to a second order
structure. We designed a novel structure for a second order
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