Download Studies on Tuning of Integrated Wave Active Filters Johan Borg LiTH-ISY-EX-3401-2003

Studies on Tuning of Integrated
Wave Active Filters
Johan Borg
LiTH-ISY-EX-3401-2003
Linköping 2003-06-05
Studies on Tuning of Integrated
Wave Active Filters
Master’s Thesis
performed at
Electronics Systems
Linköpings Universitet
by
Johan Borg
LiTH-ISY-EX-3401-2003
Supervisor: Emil Hjalmarson
Linköpings Universitet
Examiner: Professor Lars Wanhammar
Linköpings Universitet
Linköping, 2003-06-05
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Electronics Systems,
Dept. of Electrical Engineering
581 83 Linköping
2003-06-05
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Svenska/Swedish
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×
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—
LITH-ISY-EX-3401-2003
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URL för elektronisk version
http://www.ep.liu.se/exjobb/isy/2003/3401/
Titel
Studie av avstämning av integrerade aktiva vågfilter
Title
Studies on Tuning of Integrated Wave Active Filters
Författare
Author
Johan Borg
Sammanfattning
Abstract
The first part of this thesis contains a literature study of current tuning techniques for continuous-time integrated filters. These tuning methods are characterised by which quantity they measure, their dependence on certain characteristics of the input signal, or matching of components on chip. The structure of
the different tuning schemes are explained. The merits and drawbacks as well
as achieved accuracies of previous works are summarised.
The second part is a study of wave active filters (WAFs), a less common
structure for implementing active filters. In this structure the filter is realised
by simulating the forward and reflected voltage waves present in the prototype
filter. The main advantage of this is that the inherent low sensitivity of doubly
terminated ladder-filters is better preserved than in many other structures. Two
Mosfet-C realisations of Wave Active Filters have been suggested and highlevel simulations have been used to compare them to the originally proposed
implementation as well as a leapfrog implementation.
Nyckelord
Keywords
tuning, integrated filter, wave active filter, WAF
Abstract
The first part of this thesis contains a literature study of current tuning techniques for continuous-time integrated filters. These tuning methods are characterised by which quantity they measure, their dependence on certain
characteristics of the input signal, or matching of components on chip. The
structure of the different tuning schemes are explained. The merits and drawbacks as well as achieved accuracies of previous works are summarised.
The second part is a study of wave active filters (WAFs), a less common
structure for implementing active filters. In this structure the filter is realised
by simulating the forward and reflected voltage waves present in the prototype filter. The main advantage of this is that the inherent low sensitivity of
doubly terminated ladder-filters is better preserved than in many other structures. Two Mosfet-C realisations of Wave Active Filters have been suggested
and high-level simulations have been used to compare them to the originally
proposed implementation as well as a leapfrog implementation.
Table of Contents
1 Introduction
1
1.1 Background
1
1.2 Outline of this Thesis
2
1.3 Purpose of this Thesis
2
2 On-Line Tuning
3
2.1 Master-slave Frequency Control
3
2.1.1 Gm or R - only Tuning
3
2.1.2 Capacitor Charge Based Tuning
4
2.1.3 Integrator and First-Order Filter Based Tuning
7
2.1.4 Phase-Locked Filter
8
2.1.5 Phase-Locked Oscillators
10
2.2 Master-Slave Q-value Control
12
2.2.1 Phase-Locking an Integrator
12
2.2.2 Amplitude Locking Passband Gain
13
2.2.3 Envelope Based Q-value Tuning
16
2.3 True On-Line Tuning
18
2.3.1 The Correlated Tuning Loop
18
2.3.2 Orthogonal Reference Tuning
20
2.3.3 Tuning by Using Common Mode Signals
21
3 Off-Line Tuning
23
3.1 Frequency-Tuning
23
3.1.1 Step Response
23
3.1.2 Forced Oscillation
24
3.2 Combined Frequency and Q-value Tuning
24
3.2.1 Sweeping the Frequency Control Voltage
24
3.2.2 Two Reference Frequencies
24
3.2.3 Three Reference Frequencies
25
3.2.4 Isolation of Sub-Circuits
26
3.2.5 Model Matching
26
4 Wave Active Filters
29
4.1 Introduction to Wave Active Filters
29
4.2 Sensitivity
31
4.2.1 Time Constant Errors
32
4.2.2 Gain Errors
34
5 Mosfet-C Implementation of WAFs
37
5.1 Background
37
5.2 Possible Structures
38
5.3 Sensitivity to Component Errors
40
5.4 Sensitivity to OP-Amp Bandwidth Variations
41
6 Mapping of S-parameter Errors to Passive Components
45
6.1 Analytical Mapping
45
6.2 Approximate Mapping by Optimization
46
7 Tuning Strategies for Wave Active Filters
51
8 Conclusions and Future Work
53
8.1 Tuning of Continuous-Time Integrated Filters
53
8.2 Wave Active Filters
53
9 References
55
Chapter 1 – Introduction
1
1 Introduction
1.1 Background
Even though continuous-time integrated filters are usually replaced by
switched capacitor filters where feasible, many important applications
remain, such as anti-aliasing filters for high-speed data-converters and readchannel equalizers for hard-disk drives.
The main reason for using continuous-time filters is their speed. A compairable switched capacitor filter for signals in the MHz-range or higher would
require excessively high clock frequencies, with high power consumption and
clock-feedthrough as a result. Furthermore, high-performance operationalamplifiers (OP-Amps) will be required to obtain settling-times sufficiently
low for the switched capacitor circuits to reach steady state within half a
clock period.
On the other hand, the main reason for using switched capacitor filters is their
stability. Since all passive elements are realised using capacitors only, the frequency characteristics will only depend on the capacitor sizes and their relative accuracy, which are typically less than 0.1% [1], and the clock frequency.
For continuous-time filters this is not true, both capacitors, and either resistors or transconductors are used to realise the filter, the ratio of their sizes will
determine the overall frequency characteristics. Unfortunately, chip to chip
variations of RC or Gm/C can be in the order of 30% [2].
Because of this, it is usually necessary to implement some form of frequency
control, “tuning”, to ensure that the filter meets the specification.
Integrated filter design is further complicated by the fact that high performance filters are sensitive to component variations. Because of this, it is often
necessary to introduce some type of control over other parameters in the filter, in order to compensate for effects such as parasitic loads and device mismatch.
The sensitivity to component variations is also highly dependent on the structure of the filter, for example lattice, filters are generally only suitable for
crystal filters, as they are extremely dependent on element stability, while
doubly terminated LC-ladder filters are relatively insensitive to small changes
in component values.
2
Studies on Tuning of Integrated Wave Active Filters
1.2 Outline of this Thesis
•
Chapter 2 - On-Line Tuning
Chapter 3 - Off-Line Tuning
These two chapters contain the results from a literature study on the subject of tuning of continuous-time filters.
•
Chapter 4 - Wave Active Filters
A background on wave active filters as well as an initial study of their
performance in respect of component variations.
•
Chapter 5 - Mosfet-C Implementation of WAFs
Attempts at finding a Mosfet-C implementation and the performance of
the resulting candidates.
•
Chapter 6 - Mapping of S-parameter Errors to Passive Components
Further studies of the relation between filter defects in S and component
domains.
•
Chapter 7 - Tuning Strategies for Wave Active Filters
Some words on proposed tuning strategies for WAFs
•
Chapter 8 - Conclusions and Future Work
•
Chapter 9 - References
1.3 Purpose of this Thesis
The purpose of this thesis:
•
Perform a literature study of present works on tuning of continuous-time
integrated filters.
•
Study possible MOSFET-C implementations of wave active filters.
•
Investigate if it is possible map scattering-parameter errors of wave active
filters to component errors.
Chapter 2 – On-Line Tuning
3
2 On-Line Tuning
Because the parameters of integrated active filters depend on temperature,
supply-voltage and ageing, a tuning method that is active at all times (referred
to as “on-line”), is usually required. The opposite is off-line tuning, where the
filter is only tuned when inactive.
The most common way to tune an integrated filter is by using a “masterslave” tuning scheme. One or more filters are used as reference, continuously
tuned by a control circuit to meet some reference performance. The control
signals from this process can then be used for tuning the filter(s) processing
the actual input signal. One way of implementing this is shown in Fig. 1.
Since the slave filter is never measured on, the accuracy of the tuning will be
limited by the matching of the master and slave filters. Another problem with
having the reference filter and the tuning-circuit operating continuously is the
possibility of undesired signals from the tuning process leaking into the main
signal path.
Vref
master
filter
.
.
control
Vc1
.
.
.
.
.
Vin
.
slave
filter
Vcn
Vout
Figure 1: The principle of master-slave tuning
2.1 Master-slave Frequency Control
2.1.1 Gm or R - only Tuning
Where the required frequency accuracy is low, simply making sure that the
transconductances (for Gm-C filters) or resistances (for (R-)MOSFET-C) are
correct provides a simple solution. Since variations in capacitor values are not
taken into account, the accuracy of the filter after tuning will be limited by the
process variations of the capacitor values, which is usually about 10% [2].
4
Studies on Tuning of Integrated Wave Active Filters
R ext
Gm
-Vb
I
Iout
CI
Vf
Figure 2: Gm - only tuning
For example, in Fig. 2, the voltage difference Vb will make the transconductor output a current, Iout=GmVb. At the same time, there is a voltage difference
Vb over the off-chip resistor Rext, resulting in a current I=Vb/Rext. If these currents are equal, no current is going into the integrator. An incorrect transconductance Gm will cause a difference in currents, this difference will be
integrated over time, until the control-voltage Vf has changed enough to correct the Gm value.
In [3] a 7th-order equiripple lowpass filter tuneable over 30-100MHz was
built. Since it is designed for hard-disk read-channel equalizing, no data on
cut-off frequency accuracy is available, as this is secondary to the group delay
ripple.
Similarly, in [4] an elaborate scheme for tuning ratios of conductances and
time constants are presented. Maintaining these ratios is in this case necessary to ensure that the filter meets the group delay ripple specification. On the
other hand, the cut-off frequency control is mentioned only as “external”.
2.1.2 Capacitor Charge Based Tuning
A more accurate method is to replace the reference resistance with a switched
capacitor equivalent, and thereby control a Gm/C or R/C ratio directly. In theory this would look like Fig. 3, but that approach is usually not realistic, due
to the high clock frequency required if Gm and C have similar value to those
used in the slave filter (which is preferable to achieve good matching). This
can be solved by using the circuit in Fig. 4, which scales down the clock frequency a factor N, by using two currents of a ratio 1:N.
Chapter 2 – On-Line Tuning
φ1
Cm
φ2
5
φ2
CI
φ1
Gm
Vf
LP
filter
-Vb
Figure 3: Gm/C tuning by using SC-circuit
NIB
IB
Gm
-Vb
φ1 C m
φ2
φ2
φ1
CI
LP
filter
Vf
Figure 4: Improved Gm/C tuning using SC-circuit
A lowpass filter is usually required to reduce reference-signal leakage into the
slave filter. While simple, these filters can become quite large, due to the large
capacitances required for large time-constants. In some cases an off-chip
capacitor has been used [5].
Another option is to lock the time-constant directly to a reference clock, like
in Fig. 5, where the capacitor C is charged with a current determined by the
transconductor, and the peak reached during 1/2 clock cycle is compared to
the voltage Vb [6].
Vb
Tclk
Vf
Gm
Vb
φ1
Peak
Hold
G
Figure 5: Gm/C tuning by locking the time constant to the period of a
reference clock
6
Studies on Tuning of Integrated Wave Active Filters
For active R-C and similar filters, the circuit in Fig. 6 could be used, either
with Vf directly controlling a bias to the mosfet-resistances in case of a Mosfet-C filter, or through a comparator controlling a counter, in turn switching
different R or C elements in or out [8].
Here the current Vb/R through the resistance will be balanced against the current -fVbCm transferred by the switched capacitor.
CI
R
Vb
φ1
φ2
Cm
φ2
Vf
φ1
Figure 6: R-C filter version of tuning using SC-circuit
All methods discussed so far have the advantage of being very simple, and as
opposed to most other methods, the reference-signal is not required to be a
sine wave with low distortion.
Using a reference clock of a frequency considerably lower than the operation
frequency of the filter, like in Fig. 4, will reduce the problem with referencesignal leaking into the main signal path.
Accuracy will largely depend on offsets in the active components, but also on
achieving good matching with the slave filter. This may be difficult since the
structure of the master filter is fundamentally different from the slave filter.
This may result in parasitics affecting the nodes differently, with a systematic
error as result. Tracking of production spread and temperature variations are
also likely to be relatively low when these methods are used.
In [7] a 4th-order 10.7MHz bandpass filter was tuned to a frequency accuracy
of 1% by using a circuit similar to that in Fig. 4.
In [8] a 14th-order Chebyshev bandpass filter operating in the 165-505kHz
range, was tuned to an accuracy of 1% by a circuit similar to that in Fig. 3.
Here a reference-frequency well above the operating frequency of the filter
was used.
In [9] three different 78kHz active-RC filters with 5 bit binary weigthed
switchable capacitor arrays, controlled by a circuit similar to that of Fig. 6
operating in a dual-slope mode were implemented. Frequency accuracies of
5% were obtained.
Chapter 2 – On-Line Tuning
7
2.1.3 Integrator and First-Order Filter Based Tuning
An ideal Gm-C integrator will have the transfer function
Gm
H ( s ) = ------sC
(2.1)
Solving for |H(ω)|=1, we get:
Gm
ω = -------C
(2.2)
Which means that the unity gain frequency of the integrator will be
1 Gm
f = ------ -------2π C
(2.3)
As described by Fig. 7, this can be used to control the Gm/C ratio, comparing
the peak level of the reference-signal before and after it has passed through a
reference integrator. If the Gm/C ratio is correct, the output from the integrator should have the same amplitude as the input. Any difference in amplitudes
will be integrated over time by the second integrator, and the control signal Vf
changed to modify the value of Gm until the correct Gm/C ratio is obtained.
The signal Vf is then used to control the transconductances in the slave filter.
Peak
Detect
Vref
LP
filter
G
Peak
Detect
Gm
Vf
CI
C
Figure 7: Tuning using unity-gain frequency of the integrator
For a non-ideal integrator it can be shown [10] that the frequency error will be
below 0.1% if the DC-gain is larger than 40dB and the phase-error at unity
gain is smaller than 1 degree.
8
Studies on Tuning of Integrated Wave Active Filters
Envelope
Detector
Vref
.5
LP
filter
Gm
Gm
Vf
Envelope
Detector
C
Figure 8: Tuning using a degenerated integrator.
When this method is used, the input offset of the transconductor must be low
enough to keep it from saturating, since no DC loading or DC feedback exist.
One way of avoiding this problem has been proposed in [11]. In Fig. 8, the
new transconductor will simulate a resistance R=1/Gm, and the transfer function becomes:
1
H ( s ) = ----------------sC
1 + -------Gm
(2.4)
Here, instead of the unity gain, the -3dB frequency is used.
The choice of using a peak-detector or the square of the signal and low-pass
filtering the result, for measuring a signal amplitude, seems arbitrary in most
cases, but here the latter might have an advantage. This is because taking the
square of a signal with a relative amplitude of -3dB will result in an output
DC-level of half that of a 0dB input signal. A peak-detector is on the other
hand designed to preserve a linear relationship between input amplitude and
the output voltage, and will thereby produce an output of ( 1 ⁄ 2 ) times that of
a 0dB signal. In this case, when a ratio of the signals should be 3dB, implementing the attenuator after the squaring amplitude detector may improve
accuracy, since it is usually easier to implement accurate integer ratios.
This type of tuning has also been implemented in [12], [13] and [14], for tuning different circuits, but no useful experimental data is available on tuning
performance.
2.1.4 Phase-Locked Filter
The main feature of this method is that good matching between master and
slave is relatively easy to obtain, since both are filters and can be built using
similar structures.
Chapter 2 – On-Line Tuning
Vref
master
filter
9
LP
filter
Vf
Figure 9: Tuning using a phase-locked filter
In Fig. 9, a sine-wave reference-signal is used as input to the master filter.
The phase of the output signal from the filter is compared with that of the reference-signal. The phase comparison is carried out by multiplying the signals, as the DC-component of the product of two signals with the same
frequency will depend on the phase difference. If there is a 90 degree phase
difference the output will be zero. The output from the multiplier is integrated
over time, and used as the frequency control signal. This will effectively lock
the phase-shift through the filter at 90-degrees, as a different phase shift will
produce a DC output, which will be integrated until the control signal has
changed enough to correct the phase shift.
A second order lowpass filter is usually used for the master filter, as it will
have a 90 degree phase shift at its -3dB frequency. This is true even when the
slave filter is of a different type or order, because locking to a 90 degree difference usually simplifies design. Filters of higher order may also have more
than one frequency where the phase difference is 90-degrees. Thus, there is a
possibility that the tuning-circuit may converge to the wrong frequency (provided the tuning range is sufficiently large).
Other types of filters may be used, however using a filter with a ± 90 degree
phase shift at the reference-frequency usually simplifies the design. If 0 or
180 degree phase shift is used, either the quadrature component of the reference-signal or an additional 90 degree phase-shift will be required.
In some applications it might still be advantageous to use a notch-filter
instead [15], especially if the location of a zero in the transfer function is
important. When using a notch-filter as a reference, the output signal
approaches zero as the frequency of the zero in the notch-filter approaches the
frequency of the reference-signal. This will theoretically reduce the reference-signal leakage to the main signal path and reduce the size of the LP-filter in the frequency control loop.
10
Studies on Tuning of Integrated Wave Active Filters
The phase-comparator can be a major error source, as a phase-error of 1
degree will cause a frequency-tuning-error of 0.5%, if the reference-filter is a
2nd order lowpass with a Q-values of 2. Using higher Q-values will reduce
this error, but may result in reduced matching of the master and slave filters.
If the initial tuning error is large enough to make the reference-frequency fall
well inside the stop-band of the master filter, the amplitude of the input signal
to the phase-detector will be low. If the phase-detector is based on direct multiplication of the signals, the decreasing input signal amplitude will lead to a
reduction of output signal amplitude. For large tuning errors, this effect will
overtake the phase-detection and cause an overall decrease in output from the
phase-detector. Depending on the feedback loop design, this may cause convergence problems. A solution for this problem is to decrease the Q-value of
the master filter, as this will make the slope of the phase shallower, and this
make the variations in amplitude less dramatic. Alternatively, it should be
possible to avoid this problem by using a feedback loop that contains an integrator, as the sign of the phase signal will always be correct, even if the
amplitude shows inconsistencies.
In [16] an integratorless feedback loop was used with this type of phasedetector. This resulted in a requirement of Q<2 to ensure convergence over a
30% range.
In [17] a 5th-order elliptic 1.92MHz lowpass MOSFET-C filter was tuned, no
data on absolute frequency accuracy were presented, but the temperature
coefficient of the cut-off frequency is said to be 100ppm/ ° C.
In [18] an 2nd order 78kHz lowpass active-RC filter using digitally programmable current attenuators was tuned to an accuracy of 5%, of which the quantization error may account for 1-3%.
In [19] an unusually large ratio of master/slave cut-off frequencies was used,
this resulted in relatively large temperature and supply voltage dependencies
for the center frequency and Q-value.
2.1.5 Phase-Locked Oscillators
To eliminate the requirement of a low-distortion sine wave reference-signal
and the absolute accuracy of the phase-detector, phase-locking of an oscillator implemented with a structure similar to that of the slave filter, is often
used. However, in order to make sure the circuit forms a stable oscillator with
the active elements operating in their linear regions, new elements like nonlinear negative resistances, modified transconductors or limiters are usually
Chapter 2 – On-Line Tuning
11
required. These changes make a good matching to the slave filter harder to
achieve compared to a phase-locked filter. Another approach is to try to keep
a filter section oscillating by increasing the Q-value to infinity. This, however,
also tend to cause a systematic frequency error.
Vref
master
filter
LP
filter
Vf
Figure 10: Tuning using a phase-locked voltage controlled oscillator
In Fig. 10 an oscillator is formed by inserting a limiter in the feedback loop
from the output to the input of a bandpass filter, which must have a passband
gain larger than unity. The limiter will crop the peaks of the signal to some
level. This ensures that the amplitude of the input signal is low enough for the
filter to be sufficiently linear. Too high input signal amplitude will make nonlinearities in the filter significant, with a change of oscillation frequency as a
result.
When the tuning is complete, the oscillator is phase-locked to the referencesignal and any frequency error will make the phase error increase over time.
This in turn will change the DC-output from the phase-detector and adjust the
control signals for the filter. Because the phase error is the frequency error
integrated over time, no stationary frequency error will remain.
Depending on the phase-detector used, locking range may be limited to only
one octave, which is sufficiently wide to handle the tuning range of most filters. However, in some cases when this method is used with very wide-band
tuneable MOSFET-C filters, means for avoiding locking to harmonics may be
required.
In [20] a 5th-order 3.4kHz elliptic lowpass filter was implemented, with a
production spread after tuning of 0.5%, and a temperature dependence of
0.1% over the range 0-85 ° C.
In [21] a 4th-order 70MHz bandpass filter was implemented, with a systematic frequency error of 1.5% and a production-spread of 1%.
In [22] a 1MHz 2nd order active-RC using programmable capacitor arrays,
with frequency errors within 2% after tuning were implemented.
12
Studies on Tuning of Integrated Wave Active Filters
2.2 Master-Slave Q-value Control
For a pole Pk, the Q-value is defined as
Pk
Q k = – ----------------Re ( P k )
(2.5)
For biquad filters, this is directly applicable to each biquad individually, as
they implement one pair of complex conjugated poles each. When a filter of a
higher order than two is implemented in a single structure, Q-values will be
defined only for the realized poles, with no direct connection to the filter
implementation as such.
In any case, making sure that the poles of a filter doesn’t move too far from
their desired positions will be critical for ensuring that the shape of the passband remains acceptable.
The Q-values present in active filters are usually determined by a ratio
between values of similar components, like Gm1/Gm2 or C1/C2. Since the size
ratio between components of the same type is relatively insensitive to process
and temperature variations, Q-values should also be relatively insensitive and
therefore not require any tuning. This is usually true for low frequencies and
for low Q-values, where the component ratios are small and the active components are nearly ideal. At higher frequencies and larger component ratios,
nonidealities, parasitics and process variations may cause considerable deviations from the desired Q-value.
The common methods of adjusting Q-values in a filter, are either adjusting
the ratio of the component values that determines the Q-value, introducing a
controllable (positive or negative) resistance element in the circuit, or in case
of 2-stage active elements, adjusting a compensation circuit inside the element.
When the frequency and Q-value tuning are not entirely independent, the Qcontrol loop is usually made an order of magnitude slower than the frequency
control to make sure that the Q-value tuning is preformed at the correct frequency.
2.2.1 Phase-Locking an Integrator
It can be shown that a phase error in the active components of a filter will
have considerable effect on the Q-value. When a single integrator is used as
Chapter 2 – On-Line Tuning
13
master for frequency control, see section 2.1.3, this phase-error will cause the
phase difference over the integrator (after frequency-tuning) to differ from the
ideal 90-degrees. This has been used in the tuning scheme presented in
Fig. 11.
Gm
LP
filter
Vref
VQ
CI
Cm
Figure 11: Q value tuning using phase difference
Here the reference signal and the output signals are converted to logic levels
and used as inputs to an xor-gate. If the phase difference is not 90-degrees,
the output from the xor-gate will not have a 50% duty-cycle. This will cause a
non-zero average output current from the transconductor, charging or discharging the capacitor CI and thereby adjusting the control voltage VQ.
The accuracy of this method will depend on the achievable phase accuracy of
the phase-detector. It should be remembered that only phase errors caused by
nonidealities in the transconductor are measured and corrected, errors originating from inaccurate component ratios, due to process variations or parasitics, are not.
In [10] a 4MHz 6th-order elliptic lowpass filter was tuned by this method,
they claim good theoretical accuracy for the phase-detector based on comparators and a xor-gate, but no experimental data on the performance of the Qvalue tuning is presented.
2.2.2 Amplitude Locking Passband Gain
Fig. 12 shows the most common way of implementing Q-value tuning, simply using that the passband gain of a 2nd order bandpass filter will be proportional to the Q-value. If we assume that the mid-band gain is equal to the Qvalue, a too low Q-value will produce an output lower than that of the amplified reference-signal, this difference will be integrated over time, until the
control-signal VQ has changed enough to correct the Q-value. This signal is
also used to control the slave filters.
14
Studies on Tuning of Integrated Wave Active Filters
If a biquad filter is used as master in the phase-locked filter frequency-tuning
loop, a bandpass-filtered signal is usually already available in the circuit, otherwise, a separate Q-value tuning master is used.
Q
Peak
Detect
VQ
LP
filter
Vref
master
filter
Peak
Detect
Figure 12: Passband amplification based Q-tuning
Q-value tuning is often used to compensate not only for nonidealities of the
active elements, but also for component mismatches caused by parasitics and
process variations. There have been implementations with one Q-master
identical to each stage in a chain of biquads. In [23] four stages were used to
make sure that all stages were compensated correctly, instead of trying to
scale the compensation circuits.
If a frequency-tuning-error is present, the reference-frequency will not be
exactly in the center of the passband. Because of this, the gain meassured
when the reference-signal is feed through the filter will not be the passband
gain of the filter. This will result in a Q-value tuning error, since the tuningcircuit will make the meassured gain equal to the desired passband-gain, and
the actual passband gain will be forced to some different level. This error will
be approximately proportional to the Q-value, as the passband width is the
inversely proportional to the Q-value.
It has been suggested [24] that this error can be reduced (ideally eliminated)
by using the circuit in Fig. 13.
+
+
-
Vref
1/Q
master
filter
+
-
LP
filter
+
Figure 13: Improved passband amplification based tuning
VQ
Chapter 2 – On-Line Tuning
15
Here the change in VQ is calculated as
V̇ Q = µ ( V ref – V bp )V bp
(2.6)
Where Vref and Vbp are the reference-signal before and after it has passed
through the (bandpass) master filter. µ is the integrator gain.
When the tuning is complete, and no frequency error is present, both amplitude and phase will be equal. In case a frequency-tuning-error is present,
there will also be a phase shift φ trough the filter, which will make this circuit
adjust VQ until the following condition is meet:
V bp = V ref cos φ
(2.7)
This means that when the tuning is complete, the gain of the filter will be
H ( φ ) = Q cos φ
(2.8)
However, for a second order bandpass filter
ω0 s
H ( s ) = --------------------------------ω
2
2
0
s + ------s + ω 0
Q
(2.9)
the phase shift trough the filter will be
2
2
ω0 + ω
tan φ = Q ------------------ωω 0
(2.10)
Eq (2.9) and (2.10) gives
Q
H ( φ ) = i ------------------- = iQ cos φ ( cos φ + i sin φ )
tan φ + i
(2.11)
and the magnitude response as a function of the phase shift will be
H ( φ ) = Q cos φ
(2.12)
Comparing this with Eq. (2.8), we now see that the filter will ideally be tuned
to the correct Q-value, even if the reference-frequency is not in the exact
center of the passband.
This method can actually be seen as an Least Mean Square (LMS) adaptation
algorithm implementation, where the output from the master filter is used as
an approximative gradient signal.
16
Studies on Tuning of Integrated Wave Active Filters
In [24] a 10.7MHz single biquad bandpass filter with a Q-value of 20 was
manufactured and a Q-value error of 0.75% was measured (after tuning). Discrete tests of a similar circuit with a Q-value of 10 indicated that a 3% frequency error would result in an 1.1% Q-value error. If a normal amplitude
comparing Q-value tuning-circuit had been used, a 3% frequency-tuningerror at a Q-value of 10 would have resulted in a 16% Q-value error.
Fig. 14 shows another proposed method [25], which eliminates the requirement of a separate Q-tuning master filter when using a phase-locked oscillator for frequency control. According to [25] the method reduces the
sensitivity to offsets in the tuning-circuit compared to the previous method.
+
+
fctrl
VQ
-
master
filter
1/Q
LP
filter
LP
filter
Vref
Figure 14: Combined frequency and Q-tuning scheme
A 2nd order 100MHz bandpass filter with a Q-value of 20 was manufactured,
and a tuning accuracy in the order of 1% was measured.
2.2.3 Envelope Based Q-value Tuning
When a step is applied to a second order lowpass filter, the envelope of the
oscillations will be equal to the step-response from a first order low-pass filter, with a -3dB frequency of half the 2nd order filters bandwidth, as shown in
Fig. 15.
In [26] it was proposed that this may be used for tuning the Q-value of a filter
as shown in Fig. 16
Here Vref is a square wave reference-signal, with a frequency low enough to
allow the filters to reach steady state after each transition.
Chapter 2 – On-Line Tuning
17
1
V(t)
0.5
0
−0.5
−1
0
2
4
t
6
8
Figure 15: 2nd order vs 1st order lowpass filter
reference
filter
envelope
detector
(1st order)
S/H &
Vref
master
filter
(2nd order)
VQ
LP-filter
envelope
detector
Figure 16: Envelope based Q-tuning
The output signals from the two filters pass trough the envelope detectors,
which produce an output proportional to the square of their inputs. The outputs from the envelope detectors are then compared and the difference, integrated over time, used to control the Q-value of the 2nd order filter.
By controlling the sample and hold (S&H) circuit to only sample when the
filters have reached steady state, the signal leakage to the slave filters can be
reduced.
In [26] a board level test circuit was built, and Q-tuning errors of 3-7% meassured.
18
Studies on Tuning of Integrated Wave Active Filters
In [7] a 10.7MHz 4th-order biquadratic bandpass filter with Q=20 for both
biquads was implemented, with a systematic Q-tuning error of 20% and chip
to chip variations of 10%. This is attributed to offsets in the comparing
transconductor, inaccuracies in the envelope-detection and frequency-sensitivity proportional to Q.
While not as accurate as the improved amplitude locking described in 2.2.2, it
may well be comparable to the classic amplitude locking method and, if properly implemented, provide an acceptable level of accuracy.
The low frequency of the reference-signal will help reduce reference-signal
feedthrough to the slave filter, and possibly reduce the power consumption.
2.3 True On-Line Tuning
Ideally, one would want to measure the characteristics of the actual filter, like
in off-line tuning, and at the same time be able to have both tuning and signal
processing active at all times.
2.3.1 The Correlated Tuning Loop
In [27] a method for true on-line tuning of a filter is presented. It is similar to
the methods described in 2.1.4 and 2.2.2 as it tunes the filter by observing the
transfer function of the filter at a single reference-frequency. Instead of
actively providing the filter with a known input signal, and measuring amplification and phase shift at the output, these parameters are derived from the
input signal. This tuning method assumes that the input signal has sufficient
spectral contents at the reference-frequency, if this is not the case, convergence of the tuning loop can not be guaranteed.
For a linear system, the relation between the spectra at the input and output
can be written
G xy ( ω ) = G xx ( ω )H ( ω )
(2.13)
where Gxy and Gxx are the cross power and input power spectral densities,
respectively.
Chapter 2 – On-Line Tuning
19
It can be shown [27] that signals Va,Vb, Vc and Vd calculated as
t
Va =
∫ y ( u ) sin ( ω0 u )hLP ( t – u ) du
(2.14)
–∞
t
Vb =
∫ y ( u ) cos ( ω0 u )hLP ( t – u ) du
(2.15)
–∞
t
Vc =
∫ x ( u ) sin ( ω0 u )hLP ( t – u ) du
(2.16)
–∞
t
Vd =
∫ x ( u ) cos ( ω0 u )hLP ( t – u ) du
(2.17)
–∞
are orthogonal representations of the energy at the input (Vc and Vd) and output (Va and Vb). The signals will be low frequency or DC, with a bandwidth
determined by the lowpass filter hLP. It can also be shown that
V x ( t ) = V b ( t )V c ( t ) – V d ( t )V a ( t )
(2.18)
V y ( t ) = V a ( t )V c ( t ) + V b ( t )V d ( t )
(2.19)
2
2
V ref ( t ) = V c ( t ) + V d ( t )
(2.20)
will be estimates of the average real and imaginary parts of Gxy and the Gxx,
respectively. The center frequency can then be locked by using either Vx or
Vy, depending on the filter tuned, as a measure of the error in phase shift
trough the filter, and integrate this signal to create the frequency control signal. Amplitude, and thus Q-value, can similarly be created from the signal not
used for frequency-tuning, combined with Vref.
The proposed tuning system is shown in Fig. 17, where Vf (=Vx) is used for
controlling center frequency of the filter, while VQ (=Vy-Vref) is used to force
the gain to unity, in this case corresponding to a Q-value of 10.
20
Studies on Tuning of Integrated Wave Active Filters
Vf
VQ
Va
Vd
x(t)
2nd Order
Bandpass
Filter
y(t)
Vf
Vb
Vc
Va- Vc
Fox(t)
Foy(t)
Fox(t)
Foy(t)
Vc
LP
Filter
LP
Filter
LP
Filter
LP
Filter
Vd
Vc
Vb
Va
VQ
Vb- Vd
Vd
Figure 17: Tuning by using correlation of input and output signal
Here the signals Fox and Foy are the reference-signals, with a phase shift of 0 °
and 90 ° respectively, x and y are the input and output signals of the system
while Vf and VQ are frequency and Q control signals for the filter.
In [28] this tuning scheme was used for tuning an 2.5MHz 2nd order bandpass filter with a Q of 10, implemented in a 2µm CMOS process. With a full
swing input signal a frequency-tuning-error of 0.2% and a gain error of 1.1dB
was obtained.
2.3.2 Orthogonal Reference Tuning
If assumptions about the input signal, as in 2.3.1, can not be made, but the
required signal to noise ratio is low, the tuning method proposed in [29], may
be an option for true on-line tuning of the filter.
Here an approximative orthogonality is created between the reference and the
input signal by phase modulating a reference-signal with a pseudo random
sequence before adding it to the input signal, as shown in Fig. 18.
This will result in the reference-signal being spread out over a frequency
band, with the width determined by the rate of the phase modulation signal.
The output signal from the filter is then multiplied by the modulated reference-signal and its quadrature components, producing estimates of the real
and imaginary parts of the transfer function at the reference-frequency. The
result is then used to tune the filter, as described in 2.3.1.
Chapter 2 – On-Line Tuning
21
Automatic
Gain
Control
x(t)
Spreading
Sequence
Generator
Tunable
Bandpass
Filter
y(t)
Control
Loop Filter
Lowpass
Filter
Vref
Reference
Recovery
Correlators
Figure 18: Orthogonal reference tuning
A board level test circuit was built, tuning a 2nd order 10.7MHz bandpass filter with Q-value of 100. To make the theoretical accuracy of the Q-tuning
10%, a carrier to reference (C/R) ratio of 20dB, using a control loop bandwidth of 10-4 times the modulation rate was required. For the test circuit a
modulation rate of 10kHz and control loop bandwidths of 1.6Hz was
selected.
A possible application for the proposed tuning scheme would be in receivers
for wideband-FM and QPSK (of low dimensions) modulated signals. The test
circuit was inserted in the signal path of a FM broadcast receiver, and the reference-signal was virtually undetectable during listening tests when only the
monaural part of the signal were used.
2.3.3 Tuning by Using Common Mode Signals
Integrated continuous-time filters are usually implemented as differential circuits in order to improve linearity, by using differential transconductors or
operational amplifiers. If the filter was instead designed as two identical single-ended structures, with input and output signals feed differentially
22
Studies on Tuning of Integrated Wave Active Filters
between them, one could in theory have a tuning reference-signal present as a
common-mode signal in the filter [30].
Due to mismatch of the two single-ended filters, some residual reference-signal will be present in the output signal, and the input signal will have some
influence on the tuning-circuit.
In [30] a 7th-order equiripple filter using three biquads and a first order lowpass filter was designed and simulated. The tuning-circuit only measure on
the last biquad, but all three biquads are tuned based on this.
It is claimed that if the two single-ended filters are matched to 0.3%, the
dynamic range would be 50dB if the levels of the input and reference-signals
are equal, however, up to 80dB might be obtainable if the reference-signal
level is reduced.
In [31] a single biquad 60MHz lowpass filter was tuned by this method by
using phase-locking for tuning the cut-off frequency as described in 2.1.4 and
amplitude locking for Q-tuning as described in 2.2.2. The reference-signal
was added as common mode level at the input, and separated from the differential output by adding the outputs. For recovering the differential output signal a high CMR amplifier was used. For a 20mVp-p reference-signal added at
the input a residual level of 300uVp-p was present at the output, this should be
compared to the input signal range of 2Vp-p
Chapter 3 – Off-Line Tuning
23
3 Off-Line Tuning
As opposed to on-line tuning processes, like master-slave tuning, which are
active while the filter is operational, off-line tuning is only performed while
the filter is inactive, which may only be when the system is powered up,
depending on the application.
The advantage of off-line tuning is that the main filter is characterised,
instead of a reference circuit, thus, the accuracy of the tuning will no longer
be dependent on the matching of these circuits
While the methods described herein are mostly suited for off-line tuning, they
can in theory be used in a master/slave circuit. However, as the accuracy of
master-slave tuning is limited by the matching of the master and the slave,
using these methods are probably hard to justify, due to their larger power
consumption and their area overhead.
3.1 Frequency-Tuning
3.1.1 Step Response
In [32] a frequency-tuning scheme based on the step-response of the filter
was used, where the center frequency of a 16th-order 450kHz bandpass filter
was tuned to an accuracy of 0.33%. A step was applied to the input of the filter, and by using digital counters, the frequency of the resulting oscillations
was measured and the control voltage adjusted accordingly. This process was
carried out 3 times in a row, to reach the desired accuracy.
Since the chip was to be used in a time-division multiple-access (TDMA)
environment, tuning could be repeated often enough to ensure that long-term
parameter variations would not be a problem.
When implementing this method, one should remember that the oscillations
resulting from a step on the input will have a frequency deviating slightly
from the center frequency of the filter.
24
Studies on Tuning of Integrated Wave Active Filters
3.1.2 Forced Oscillation
Another, not very successful method, was proposed in [33]. A 250MHz 8thorder biquad R-C filter was forced to oscillate by changing the gain of one
amplifier in each biquad. The oscillation-frequency was measured using digital counters.
This resulted in a frequency-dependent systematic error of 5-10%. This is a
larger shift than can be accounted for by the change of Q-value when the filter
was forced to oscillate. One possible explanation might be the nonlinear
effects encountered when the oscillation is limited by the linear range of the
filter.
3.2 Combined Frequency and Q-value Tuning
3.2.1 Sweeping the Frequency Control Voltage
In [34] tuning by applying a (slow) triangular wave at the frequency control
input of the filter was proposed. A constant frequency reference-signal is used
as the input, and the resulting amplitude-variations of the output signal are
observed and used to tune the filter. This method is only applicable for highQ filters, with a well-defined peak in the amplitude-response.
This method is implemented by sampling the control voltage when the filter
amplification passes a level slightly below the peak level, once for rising control voltage and once for falling control voltage, and using the average of
these voltages for controlling the filter. It is also possible to use the peak output amplitude to tune the filter Q-value by the method described in 2.2.2.
In [35] this method was tested in an off-line-configuration for a single biquad
bandpass filter tuneable over 105-120MHz, with frequency-tuning-errors
below 0.3% for Q-values ranging from 34 to 83.
3.2.2 Two Reference Frequencies
In [36] a tuning scheme based on using a phase-locked VCO with a frequency-divider controlled by the tuning-circuit, producing 2 frequencies N+1
and N-1 times the reference-frequency was proposed. N/2 is approximately
equal to the desired Q-value, and N times the reference-frequency is the
desired center frequency of the filter to be tuned. A second order filter with
bandpass and lowpass outputs are assumed.
Chapter 3 – Off-Line Tuning
25
|H BP(ω )|
3dB
ω
0
-45
-90
-135
-180
ω
arg(H LP(ω ))
Figure 19: Phase-frequency relation of a 2nd order bandpass filter
As seen in Fig. 19, the 3dB frequencies of the bandpass filter will correspond
to phase shifts of -45 and -135 degrees in the lowpass filter. These phaseshifts are measured, and the frequency-tuning loop is designed to converge to:
φ ( ( N + 1 )ω r ) + φ ( ( N – 1 )ω r ) = – 180°
(3.1)
and the Q-tuning loop should converge to:
φ ( ( N + 1 )ω r ) – φ ( ( N – 1 )ω r ) = – 90°
(3.2)
It should be remembered that the statements that Q is equal to N/2, and that f0
equal to N times the reference-frequency are only approximately true, for
N<10 (Q<5) the errors will be larger than 0.5%.
It has also been suggested [37] that this method may be used to tune the individual circuits in a filter built from a chain of biquads.
3.2.3 Three Reference Frequencies
In [38] a tuning scheme similar to 3.2.2 was proposed, but in this case three
frequencies (N-1,N,N+1)ωref are used, with the signal attenuated by a factor
of two when Νωref is being generated. The center frequency of the filter is
tuned to make the output amplitude from (Ν−1)ωref and (Ν+1)ωref equal, and
the Q-value is tuned to make the amplitude from Νωref equal to that of one of
the other reference-signals, locking the -6dB bandwidth to 2ωref.
26
Studies on Tuning of Integrated Wave Active Filters
A second order 200MHz bandpass filter with a desired Q-value of 28.6 was
manufactured, and a frequency-tuning-error of 0.25% and a Q tuning error of
3% was measured.
3.2.4 Isolation of Sub-Circuits
If the shape of the passband is important, tuning center frequency and Qvalue of the filter may not be enough. One approach is to isolate sub-circuits
in the filter and tune them individually.
In [39] tuning of a leapfrog filter by isolating resonant loops in the filters, and
separately measuring their resonance frequencies was proposed. The parts of
the filter that are not part of a resonant loop may either be reconnected to
form one, or they may be tuned by the methods described in 2.1.3. Another
approach is to isolate the filter completely into first order sections, and apply
the method from 2.1.3 to each part individually.
In [40] a 6th-order narrow band Chebyshev filter was tuned one resonator at a
time, by shunting the others to ground. A frequency-tuning-error of 3% was
measured, which is suggested to be caused by nonideal characteristics of the
phase-detector used.
In [41] a 4th-order, 21.4MHz butterworth filter was tuned by isolating one
resonator at a time, and employing the tuning schemes described in 2.1.4 and
2.2.2 for frequency and Q-tuning, respectively. They obtained a center frequency accuracy of 0.014%. Here a mixed-signal implementation of the tuning-circuit was used, where a D-type flip-flop replaced the multiplier as
phase-detector, and a successive approximation scheme controlling current
DACs replaced the integrator.
3.2.5 Model Matching
In [42] the use of a model-matching algorithm for tuning continuous-time
integrated filters is proposed. Model-matching algorithms in general are originate from control theory, where they are used to create a model of a simulation model of a physical system, by observing input and output signals only.
Ideally this type of method can tune the position of all the poles and zeros in
the filters.
Chapter 3 – Off-Line Tuning
27
In this case the least mean square (LMS) algorithm is used, where the coefficients of the filter are updated by
b˙n ( t ) = 2µe ( t )φ b ( t )
(3.3)
n
where e(t) is the difference between actual and ideal output from the filter,
and φ b n ( t ) is the gradient signal. The gradient signal is defined as the derivative of the output signal with respect to parameter bn, thus, if both e(t) and
φ b n ( t ) are positive at a given instant, the output signal at this instant is too
low, but if parameter bn had been larger, the output would have been higher,
so increasing bn, as indicated by e ( t )φ b n ( t ) will reduce the error. Normally
the product of the gradient and the error signal is used, but in this implementation e ( t )sign ( φ b n ( t ) ) is used in order to simplify the multiplication circuit.
Linear time-invariant systems, such as ideal filters, can be described by a
state-space representation
sX ( s ) = AX ( s ) + bU ( s )
T
Y ( s ) = c X ( s ) + dU ( s )
(3.4)
where u(t) is the input signal, y(t) the output, and xi(t) the internal states of
the filter.
Filters for generating gradient signals (gradient filters) can then be derived as
T
A grad = A b grad = c
c grad = b
T
T
(3.5)
d grad = d
The gradient for Aij can be found from the state xi(t) in a gradient filter with
the state xj(t) in the main filter as input, similarly, the gradient for bi is found
as the state xi in a gradient filter with u(t) as input signal. The gradients for c
and d are the states of the main filter and the input signal, respectively.
Depending on the filter structure used, the tunable parameters of the filter can
be found from more or less simple relations to the parameters of the statespace description. In the article an orthonormal ladder filter was used, for
which the coefficients of the filters are found directly in A and b.
If only one parameter is being tuned at a time, only one gradient filter is
required, which takes its input from different points in the main filter, depending on which gradient signal is being generated.
28
Studies on Tuning of Integrated Wave Active Filters
The proposed tuning-circuit is shown in Fig. 20 (a), the predetermined input
is generated by a pseudo random number generator followed by a digital to
analog converter (DAC). The reference-signal is generated by another DAC,
from a table of precalculated values.
Reference
Signal
Generator
δ n(t)
Ideal
Reference
Filter
(Continuous)
+
Predetermined
Input
u(t)
Tunable
Filter
Adaptive
Tuning
Algorithm
δ(t)
+
y(t)
Predetermied
Input
e(t)
(a)
Tunable
Filter
u(t)
(Digital)
Adaptive
Tuning
Algorithm
yn (t)
e(t)
(b)
Figure 20: Tuning by LMS model matching
This table of precalculated values is generated as shown by Fig. 20 (b). Here
the desired continuous-time filter is simulated, and a digital filter tuned to
minimise the difference between the continuous and discrete filter outputs.
When the tuning is complete, the output from the digital filter can be saved
and used for tuning the real continuous-time filter.
A working discrete tuning-circuit, tuning an integrated filter, was constructed,
but no performance measures other than time to tune the filer, are given.
Use of a dithered linear search algorithm for tuning filters has recently been
proposed [43], eliminating the need for large gradient filters.
Adaptive tuning techniques can in theory also be used for tuning a filter while
it is processing signals. This is done by implementing one more identical filter, which is first tuned by an other method. This second filter is then feed the
same input as the main filter, and the adaptive algorithm is used to tune the
main filter until the output signals are identical. If the input signal has sufficient spectral contents, it would in theory be possible to tune the filter perfectly using this method, as it does not depend on the matching of the filters.
Chapter 4 – Wave Active Filters
29
4 Wave Active Filters
4.1 Introduction to Wave Active Filters
Wave active filters (WAF) was first proposed in [44], in an attempt to find an
active filter structure with the same insensitivity to coefficient errors as wave
digital filters (WDF). Instead of simulating passive filter components, as in
gyrator-C filters, or node voltages as in leapfrog filters, the filter is described
by the forward and reflected voltage waves.
I2
I1
A1
A2
R1
V1
N
B1
R2
V2
B2
Figure 21: Generic two-port
Starting from the generic two-port N in Fig. 21 (with port resistances Ri,
i=1..2), incident waves A and reflected voltage waves B are defined as
Ai = Vi + Ri Ii
(4.1)
Bi = Vi – Ri Ii
Although different port resistances are possible, they will be assumed to be
equal in all cases discussed here.
The relationship between A and B is described by the scattering matrix S as:
B1
B2
= S
A1
A2
(4.2)
The basic element when building wave active filters is the wave equivalent of
a series inductor L, which can be shown to have the scattering matrix S:
1
S = -------------- sτ 1
1 + sτ 1 sτ
(4.3)
with L=2Rτ, where R is the common port resistance. Τhis can be interpreted
as a lowpass filter from input to reflected wave signal, and a complementary
high pass filter for the transmitted wave signal. This functionality may be
implemented using the circuit in Fig. 22.
30
Studies on Tuning of Integrated Wave Active Filters
A1
B1
A2
1
1
B2
Figure 22: A simple implementation of the wave two-ports used in WAFs
As shown in Fig. 23, series and parallel connected inductors and capacitors
can be created from this element by swapping outputs or inverting signals.
More complex elements like parallel series resonators and series connected
parallel resonators can also be realised from these blocks.
L=2Rτ
A1
A2
τ
B1
B2
C=τ /2R
A1
A2
τ
B1
L=Rτ /2
B2
A1 -1
A2
τ
-1 B
2
B1
C=2τ /R
A1
-1
A2
τ
-1 B2
B1
L=2Rτ1
A1
C= τ2 /2R
B1
τ1
τ2
B2
A2
-1
C=2 τ1 /R
A1
L=Rτ2 /2
B2
B1
τ1
τ2
-1
A2
Figure 23: Wave two-port equivalents of passive components
Chapter 4 – Wave Active Filters
31
Terminating one port of the two-port adaptor with the resistance R is equivalent to having no reflected signal from B to A at that port. The output voltage
would be directly available on port B. Similarly, connecting one port to a
source with impedance R, simply means feeding the signal directly into A.
For example, the Chebyshev and Cauer filters, both of the 5th order, shown in
Fig. 24,can be realized as active wave filters according to Fig. 25
VIN U
L5
L3
L1
R
C2
C4
R
VOUT
R
VOUT
(a)
R
L5
L3
L1
C2
C4
L2
L4
VIN U
(b)
Figure 24: 5th order Chebyshev (a) and Cauer (b) lp-filters.
VIN
V'
IN
τ1
τ3
-1
-1
τ2
V'
τ5
τ4
VOUT
OUT
-1
-1
(a)
-1
-1
VIN
V'
IN
τL1
V'
OUT
τL2
-1
τL4
-1
τC2
τL3
τC4
τL5
VOUT
(b)
Figure 25: WAF realisation of 5th order Chebyshev (a) and Cauer (b) filters
4.2 Sensitivity
The performance of wave active filter implementations presented so far has
been worse than expected [45], and it was suspected that this might have been
due to lack of reciprocity in the wave two-ports, caused by unavoidable component variations.
Reciprocity basically means that a two-port has the same transfer-characteristics in both directions, when the (possibly different) port impedances has
been accounted for.
32
Studies on Tuning of Integrated Wave Active Filters
For a wave two-port where both ports have the same port resistance, reciprocity simply means that the transfer function from port A1 to port B2 is identical
to the transfer function from port A2 to port B1, or expressed from the scattering parameters: s12=s21 [46].
In order to investigate this, the two 5th order wave active filters shown in
Fig. 25 (based on Chebyshev and Cauer lowpass filters), were simulated with
different types of errors introduced.
4.2.1 Time Constant Errors
The time constant errors were created by replacing the scattering matrix S
from Eq (4.3), describing the ideal wave two-port for an inductor, with:
S =
sτe 1
1
------------------- -------------------1 + sτe 1 1 + sτe 2
sτe 4
1
-------------------- ------------------1 + sτe 3 1 + sτe 4
(4.4)
where e1..4 are error parameters, with en=1 when no error is present.
Randomly distributed errors in the range 0.99 to 1.01 were used, 10000 sets
of parameters were tested, and the largest and smallest absolute values of the
amplitude responses were plotted for 1000 frequencies in the range 0 to 2. In
the graphs presented the curve in the middle represents the ideal frequency
response. These and all simulations in the following chapters were performed
using MatLabTM.
Fig. 26 shows the result when e1..4 are allowed to vary independently.
In Fig. 27 relations e3=e2 and e4=e1 between the errors in one two-port are
maintained, as this ensures that reciprocity[46] is preserved for all the components derived as shown in Fig. 23.
In Fig. 28, the all errors in a two-port are equal.
Chapter 4 – Wave Active Filters
33
Cauer
1
1
0.8
0.8
|H(w)|
|H(w)|
Chebyshev
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
w
0
2
0
1
w
2
Figure 26: Frequency response for the Chebyshev and Cauer filters with
independent time-constant errors
Cauer
1
1
0.8
0.8
|H(w)|
|H(w)|
Chebyshev
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
w
2
0
0
1
w
Figure 27: Frequency response for the Chebyshev and Cauer filters with
reciprocity preserving time-constant errors.
2
34
Studies on Tuning of Integrated Wave Active Filters
Cauer
1
1
0.8
0.8
|H(w)|
|H(w)|
Chebyshev
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
w
2
0
0
1
w
2
Figure 28: Frequency response for the Chebyshev and Cauer filters with
equal time-constant errors
While there is a marginal positive effect of maintaining reciprocity, this
change is highly marginal and hardly visible from the plots alone. From the
very marginal result of reducing the number of error-sources by a factor of
two, one could suspect that there are other ratios that are more important to
maintain. Some attempts at finding such ratios were made, with limited success, as the results seemed to depend on which type of component, which filter structure and where in the filter the component was.
If all the added errors are kept equal in each wave two-port, any errors can be
mapped directly to component errors in the LC-filters, with the expected low
sensitivity as a result.
4.2.2 Gain Errors
In order to evaluate the effect of gain errors in the wave two-ports, similar
tests as in 4.2.1 were conducted, this time the scattering matrix
sτe 1
------------1
+ sτ
S =
e3
------------1 + sτ
e2
------------1 + sτ
sτe 4
------------1 + sτ
was used, where e1..4 are the error terms, en=1 when no error is present.
(4.5)
Chapter 4 – Wave Active Filters
35
The same number of parameters was tested (10000) and error range (0.99 to
1.01) was used, with the largest and smallest absolute value plotted for 1000
frequencies.
Fig. 29 shows the result when e1..4 are allowed to vary independently.
In Fig. 30 relations e3=e2 and e4=e1 between the errors in one two-port are
maintained, in order to ensure that reciprocity is preserved.
Chebyshev
Cauer
1
1
|H(w)|
1.5
|H(w)|
1.5
0.5
0
0.5
0
1
w
0
2
0
1
w
2
Figure 29: Frequency response for the Chebyshev and Cauer filters with
independent gain errors
Chebyshev
Cauer
1
1
|H(w)|
1.5
|H(w)|
1.5
0.5
0
0.5
0
1
w
2
0
0
1
w
Figure 30: Frequency response for the Chebyshev and Cauer filters with
reciprocity preserving gain errors.
2
36
Studies on Tuning of Integrated Wave Active Filters
The conclusions from 4.2.1 applies here too, only a very marginal improvement was seen from maintaining reciprocity, and the observed component
dependency of which, if any, relations should be maintained to reduce the
effect of errors, seemed to be roughly the same.
However, one should remember that the even distributions of random errors
used in this and previous section, combined with plots of min and max, only
give an idea about the worst case effects of errors, which may be highly pessimistic if some errors interact strongly. However, it does give a rough idea
about the relative importance of the different errors.
Chapter 5 – Mosfet-C Implementation of WAFs
37
5 Mosfet-C Implementation of WAFs
5.1 Background
The chip to chip absolute variations of transconductances, capacitances and
resistances are considerable. This results in a cut-off frequency uncertainty in
the order of 10-30%, if no correction is applied.
Fortunately, it is usually rather straight-forward to measure these variations
and adjust the filter accordingly, using the techniques described in chapter 2
and 3, improving the final accuracy an order of magnitude, or more.
However, this does require that it is somehow possible to control either resistances, conductances or capacitances, depending on the implementation.
In the case of Gm-C filters this is straight-forward, as the time-constants in
the filter will be determined by capacitances and the output transconductance
of the active elements. The transconductances are in turn determined by a
bias voltage which can easily be changed.
For active-RC filters the time-constant control is usually implemented by
realizing resistances with mosfet transistors working in the triode region.
Unfortunately these resistances are not linear. This problem is be reduced by
only implementing part of each resistance as mosfet transistor, and the rest as
a passive resistance in series. By connecting the active part closest to the OPAmp input, the voltage amplitudes over it will be low, with improved overall
linearity as a result.
Wave active filters implemented as described in the previous chapter lack any
such low-voltage node, making this type of control less useful. An alternative
control-strategy is to have a bank of discrete valued components switched in
or out to obtain the desired time-constant.
Based on this, it would be interesting to create a R-mosfet-C implementation
of the wave two-port, where all resistors are connected to a virtual ground
node at the input of an OP-Amps. In the study only structures suitable for differential implementations were considered.
38
Studies on Tuning of Integrated Wave Active Filters
5.2 Possible Structures
In an attempt to find a suitable structure, the signal flow graph of a wave twoport was transformed in various ways, the resulting graphs are shown in
Fig. 31.
A1
A1
B1
1/st
st/(1+st)
1/st
st/(1+st)
B1
A2
B2
A1
B2
A2
1/st
B1
1/(1+st)
A2
B2
A2
(A)
1/st
B2
(B)
(C)
A2
B1
A1
st/(1+st)
st/(1+st)
1/st
st/(1+st)
1/st
B1
1/st
1/st
A1
B2
A2
B1
(D)
B2
A1
(E)
(F)
B1
A1
A1
B2
B2
1/(st+1)
1/(st+1)
st/(1+st)
1/(1+st)
A1
1/st
1/st
A2
B1
A2
A2
B1
(G)
B2 (I)
(H)
A1
B1
B1
A1
A1
B1
1/st
1/(st+1)
1/(st+1)
1/st
st/(st+1)
1/(st+1)
B2
A2
A2
B2
B2
A2
(L)
(J)
(K)
Figure 31: Signal flow graphs for the basic wave two-port
These were mapped to differential mosfet-C structures, the resulting circuits
for Fig. 31 C, A and F are shown in Fig. 32, 33 and 34, respectively. All the
other structures will in fact result in a circuit very similar to one of these.
The values of all resistances and capacitances connected to the inputs of one
OP-Amp are equal. The time-constant of the resulting wave two-port is
τ=RC.
Chapter 5 – Mosfet-C Implementation of WAFs
39
+A2
+A1
+A1
-B1 +A2
-B2
-A1
+B1 -A2
+B2
-A2
-A1
Figure 32: Two OP-Amp realisation of the wave two-port (C)
+A1
-B1
+B1
-A2
-A1
+A1
-A1
+A2
+A2
-B2
+B2
-A2
Figure 33: Tree OP-Amp realisation of the wave two-port (A)
+A1
+B2
-A1
-B2
+A2
+B1
-A2
-B1
Figure 34: Four OP-Amp realisation of the wave two-port (F)
40
Studies on Tuning of Integrated Wave Active Filters
5.3 Sensitivity to Component Errors
In order to evaluate the sensitivity to component value errors for the structures in section 5.2, monte-carlo analysis based on circuit level simulations
was performed on an 5th order Cauer filter. The filter used is the pi-type
equivalent of the t-mode Cauer filter used earlier. Four different active-RC
implementations were examined, three wave active and one leapfrog implementation (for comparison).
The first wave-active implementations used the circuit in Fig. 22, with the
unity gain buffers implemented as single-ended OP-Amps with the output
connected to the inverting input. The second WAF used the two OP-Amp
active-RC implementation of Fig. 32. The last WAF was implemented using
the three OP-Amp implementation of Fig. 33, where only a single time-constant is used.
Finding realistic figures of component variations within a chip proved difficult; according to [1] a matching of 0.01% can be achieved for untrimmed
capacitors on the same chip. However, this is the matching between identical
components. No useful figures on accuracy of non-integer ratioed components were found, for the simulations a matching error with a standard deviation of 0.1% was used, as this seemed to be a reasonable value.
In all the analysed filter structures it is possible to make all resistances equal,
at the expense of capacitor ratios, but according to [1], this seems to be preferable, as the achievable matching of equal sized resistors was claimed to be
in the order of 0.1%.
The results are shown in Fig. 35. Dashed lines represents the 5th and 95th
percentiles when only resistor errors are present, dotted lines corresponds to
capacitor errors, and solid lines the combined errors.
One should note that the buffer-based wave active filter is clearly superior in
this respect. The 3 OP-Amp/two-port (single time-constant) implementation
have similar sensitivity to capacitor variations, which makes it relatively well
suited for MOSFET-C implementation, if adequate tuning-circuitry is implemented to correct the resistance values.
Chapter 5 – Mosfet-C Implementation of WAFs
41
WAF, using buffers
WAF, 2 OP−Amps/two−port
1
1
|H(w)|
1.5
|H(w)|
1.5
0.5
0
0.5
0
0
1
2
w
WAF, 3 OP−Amps/two−port
1
1
|H(w)|
1.5
|H(w)|
1.5
0.5
0
0
1
w
leapfrog
2
0.5
0
1
2
0
1
w
w
Figure 35: Component error sensitivity of 5th order Cauer filters
0
2
5.4 Sensitivity to OP-Amp Bandwidth Variations
In the previous chapter ideal OP-Amps have been assumed, however, in an
actual implementation the presence and location of poles and zeros in the
transfer function of the OP-Amp will influence the transfer function of the filter.
In this chapter an OP-Amp based on example 5.7 in [47] is used, which has a
transfer function of
s
A 0  1 + ------

ω z
H ( s ) = ------------------------------------------------s
s 
 1 + -------- 1 + ---------

ω p2
ω p1 
(5.1)
where ωz=120MHz, ωp1=4.2kHz, ωp2=143MHz and A0=80dB. This results
in an unity gain frequency of about 100MHz. These values are later scaled to
42
Studies on Tuning of Integrated Wave Active Filters
obtain unity gain frequencies suitable for the normalized frequency range
used herein.
During the initial tests with non-ideal OP-Amps it was determined that the
MOSFET-C implementations require the unity-gain frequency of the OPAmps to be about 2.5 times higher than for the unity gain buffer implementation.
Because of this, two different unity-gain frequencies are used (20 and 50
times the cut-off frequency of the filter), in order to make the comparison of
sensitivity to pole/zero location variations later more meaningful.
The results can be seen as part of Fig. 36, where the solid line represents the
ideal frequency response, while the dashed line represents the frequency
response when the nonideal OP-Amps are used.
The large difference between ideal and actual frequency response is no surprise, as ideal OP-Amps were assumed when all filters was designed.
If, on the other hand, the location of the poles and zeros of the OP-Amps are
known when the filter is designed, it is possible to adjust the filter to restore
the desired transfer function. Unfortunately, in real implementations only
approximate values are available, as pole/zero locations depend on parasitic
capacitances and the loading of the OP-Amp.
The actual adjustment was performed by applying a minimax optimization to
the filters, where the error function to be minimized was calculated as the
maximum of the largest absolute differences of the transfer functions in the
passband, and the largest violation of the minimum stop-band attenuation.
In the buffer and 2-OP-Amp implementations of the wave-active filter the two
time constants of each wave two-port was changed independently. In the 3OP-Amp implementation only one time constant is available, so an additional
gain parameter equal for both outputs was introduced.
For the leapfrog filter all the capacitors in the implementation (not only the
time constants from the original Cauer filter) were changed independently.
It was necessary to introduce these additional degrees of freedom in order to
be able to restore the passband shape. Still, only the attenuation of the stopband, not the exact shape, was preserved. One may note that in all cases the
zeros have been moved away from the imaginary axis.
Chapter 5 – Mosfet-C Implementation of WAFs
WAF, using buffers
1.5
1
|H(w)|
|H(w)|
1.5
0.5
0
43
1
0.5
0
0
1
2
w
WAF, 3 OP−Amps/two−port
1
1
|H(w)|
1.5
|H(w)|
1.5
0.5
0
WAF, 2 OP−Amps/two−port
0
1
w
leapfrog
2
0
1
w
2
0.5
0
1
w
2
0
Figure 36: Effects and compensation of nonideal OP-Amps
The result is shown in Fig. 36, dashed lines represents the original filter using
nonideal OP-Amps, dotted lines the desired frequency response, and the solid
lines the response of the filter after component values have been adjusted to
compensate for the OP-Amps.
The adjusted filters showed marginally higher sensitivity to component errors
in tests performed in the same manner as in section 5.3.
In an attempt to study the effects of differences between the pole/zero locations used when the filter was adjusted, and the real pole/zero locations of the
OP-Amps, the filters were simulated as described above, this time with normally distributed variations with a standard deviation of 10% added independently to the position of the poles and zero to all the OP-Amps.
44
Studies on Tuning of Integrated Wave Active Filters
WAF, using buffers
WAF, 2 OP−Amps/two−port
1
1
|H(w)|
1.5
|H(w)|
1.5
0.5
0
0.5
0
0
1
2
w
WAF, 3 OP−Amps/two−port
1
1
|H(w)|
1.5
|H(w)|
1.5
0.5
0
0
1
w
leapfrog
2
0
1
w
2
0.5
0
1
w
2
0
Figure 37: Pole/zero position sensitivity of 5th order Cauer filters
The results are shown in Fig. 37, where the 95th percentiles from 5000 simulations are shown. One should remember that in this case, the result will be a
distribution around the frequency response of the OP-Amp bandwidth-corrected filters, which have stop-bands with different shapes than the original
filter.
Considering the OP-Amps in the buffer implementation have a 2.5 times
lower unity gain frequency than the OP-Amps in the other implementations,
this implementation performs very well, with no peaking at the end of the
passband, and only marginal cut-off-frequency change. The leapfrog filter on
the other hand must also be considered relatively well behaved, considering
the magnitude of the uncompensated error.
One should also remember that the cut-off frequency-tuning will compensate
for the cut-off frequency inaccuracy, while filters suffering from excessive
passband ripple usually implements a Q-value tuning-circuit for suppressing
these changes.
Chapter 6 – Mapping of S-parameter Errors to Passive Components
45
6 Mapping of S-parameter Errors to Passive
Components
In order to better understand why a specific error in the scattering matrix
affects the transfer function the way it does, attempts where made at deriving
a method for mapping these errors to changes in the original LC-filter.
6.1 Analytical Mapping
The first attempt was to find an analytical expression, by inserting gain and
time-constant errors into the scattering matrix corresponding to the wave twoport description of a series inductor.
This proved to be non-trivial, for time constant errors of the same type used in
section 4.2.1, recalculating the scattering matrix (with error terms included)
into the admittance-matrix Y yields:
e 1 sτ 


2  1 + ------------------


1 + e 4 sτ
1
= ---- ------------------------------------------------------------------------------------------------------------------------- – 1

R
e 1 sτ ( 2 + ( 2e 1 + e 4 )sτ )
1
 1 – ------------------------------------------------
- + ------------------------------------------------------1
e
1
e
(
+
sτ
)
(
1
+
e
sτ
)
(
+
sτ
)
(
1
+
e
sτ
)


2
3
1
4
(6.1)
2
1
1
Y 12 = ---- ------------------------- ------------------------------------------------------------------------------------------------------------------------e 1 sτ ( 2 + ( 2e 1 + e 4 )sτ )
R ( 1 + e 2 sτ )
1
1 – -------------------------------------------------- + ------------------------------------------------------( 1 + e 2 sτ ) ( 1 + e 3 sτ )
( 1 + e 1 sτ ) ( 1 + e 4 sτ )
(6.2)
1
1
2
Y 21 = ---- ------------------------- ------------------------------------------------------------------------------------------------------------------------R ( 1 + e 3 sτ )
e 1 sτ ( 2 + ( 2e 1 + e 4 )sτ )
1
1 – -------------------------------------------------- + ------------------------------------------------------( 1 + e 2 sτ ) ( 1 + e 3 sτ )
( 1 + e 1 sτ ) ( 1 + e 4 sτ )
(6.3)
1


2  2 + --------------------



1
e 1 sτ
+
1
= ---- ------------------------------------------------------------------------------------------------------------------------- – 1

R
e 1 sτ ( 2 + ( 2e 1 + e 4 )sτ )
1
 1 – ------------------------------------------------
- + ------------------------------------------------------( 1 + e 1 sτ ) ( 1 + e 4 sτ )
 ( 1 + e 2 sτ ) ( 1 + e 3 sτ )

(6.4)
Y 11
Y 22
The gain errors from 4.2.2 result in:
46
Studies on Tuning of Integrated Wave Active Filters
1 e 2 e 3 – ( ( e 1 – 1 )sτ – 1 ) ( 1 + ( 1 + e 4 )sτ )
Y 11 = ---- ----------------------------------------------------------------------------------------------R ( ( e 1 + 1 )sτ + 1 ) ( 1 + ( 1 + e 4 )sτ ) – e 2 e 3
(6.5)
2e 2 ( 1 + sτ )
1
Y 12 = ---- --------------------------------------------------------------------------------------------R e 2 e 3 – ( ( e 1 + 1 )sτ + 1 ) ( 1 + ( 1 + e 4 )sτ )
(6.6)
2e 3 ( 1 + sτ )
1
Y 21 = ---- --------------------------------------------------------------------------------------------R e 2 e 3 – ( ( e 1 + 1 )sτ + 1 ) ( 1 + ( 1 + e 4 )sτ )
(6.7)
1 e 2 e 3 – ( ( e 1 + 1 )sτ + 1 ) ( ( e 4 – 1 )sτ – 1 )
Y 22 = ---- ----------------------------------------------------------------------------------------------R ( ( e 1 + 1 )sτ + 1 ) ( 1 + ( 1 + e 4 )sτ ) – e 2 e 3
(6.8)
While it is theoretically possible to realise any reciprocal two-port from its
impedance or admittance matrix description, the complexity and high orders
of the expressions will in this case result in complex networks. Since the purpose of this study was to better understand the mapping of errors, not finding
an exact equivalent, the attempts to find an analytical equivalent was abandoned.
6.2 Approximate Mapping by Optimization
In an attempt to find an approximative passive component equivalent, different types of networks based on doubly terminated LC filters were used. These
networks were created by adding more components to a LC-filter identical to
the filter the WAF being modelled was first based on.
The transfer function of this new network was compared to that of a WAF, in
which an error had been inserted in one of the two-ports. Minimax optimization was used to make the absolute value of transfer functions identical for a
large number of frequencies, by adjusting the component values in the network based on a LC-filter.
A number of different structures for modelling the errors were tried, some of
which will be described here.
The simplest structure just included resistors in series with every inductor and
conductances added in parallel with each series resonant circuit. The initial
values for the new components were selected to make it identical to the original LC-filter, that is, both series resitances and parallel conductances set to
zero. For the optimization the L and C values were constrained to be larger
than or equal to zero, while resistances and conductances values were left
unbounded.
Chapter 6 – Mapping of S-parameter Errors to Passive Components
47
The error function ε minimized by the optimization was computed as
1
ε = ----- max H 0 ( ω ) – H m ( ω )
ε0
(6.9)
Where H0 is the desired transfer function, Hm is the transfer function of the
model filter and ε0 is the initial error, used in consistent results from convergence limits for different magnitudes of errors in the wave active filter. The
transfer functions was calculated for 1000 frequencies in the range of 0 to 2.
The same Cauer filter as in 4.2.1 and 4.2.2 was used. The corresponding filter
built as described above is shown in Fig. 38.
R'
S
L3
L1
L4
L2
VIN U
G1
L5
R3
G3
R'
L
R2
R4
C2
C4
VOUT
Figure 38: Simple LC-filter based network for modelling WAF errors
Unfortunately, this model fail to produce good convergence in many cases.
This is not very surprising, as the nonideal wave two-ports introduce additional poles and zeros, while this model preserves the order of the filter.
Even in the cases where the optimization converge to good solutions, it tends
to generate different sets of component values which yields the same transfer
function, even for small changes in magnitude of the active error parameter.
Fig. 39 shows how the component values of the model filter changes, when
the time-constant part of S22 of the wave two-port that implements C2 of the
original filter is multiplied by 1+[-16,-8,-4,-2, 2, 4,8,16]/1000 (G1 and G3 are
the unmarked curves which are very close to zero). For example, one can see
that L1 increases when this time-constant is decreased, but remains constant if
the time-constant is increased. It is also interesting to note that C2 change less
than most of the other component values, as this is the component implemented by the wave two-port where the error is introduced.
This is one of the cases when the model works best, with good matching of
the transfer functions, with most values following some trends and few cases
where parameters have converged towards different solutions.
48
Studies on Tuning of Integrated Wave Active Filters
1.6
L4
1.4
L3
component value
1.2
RS
C
L12
R
1
L
L1
0.8
0.6
C4
L2
L
0.4
5
0.2
R2
R4
0
−0.2
−2
−1.5
R2
−1
−0.5
0
error
0.5
1
1.5
2
Figure 39: Results from modelling of WAF errors using the simple model
Another model was created by changing the components in the original filter
into the nets shown in Fig. 40a and Fig. 40b, the former used for inductors
and parallel resonant circuits, while the later was used for capacitors and
series resonant circuits. Source and termination resistances were variable during the optimization, but no additional components were added in series or
parallel to them.
The corresponding filter is shown in Fig. 41.
L
R
C
(a)
R
L
C
(b)
Figure 40: Component replacements for the extended model
Chapter 6 – Mapping of S-parameter Errors to Passive Components
R
R1
L1
C1
VIN U
L3
L2
49
L5
R3
C3
L4
C2
C4
R2
R4
R5
C5
R
VOUT
Figure 41: Extended LC-filter based network for modelling WAF errors
This model did converge better than the previous, which is reasonable, as the
order of the model-filter has increased. However, the problem with the optimization converging to different solutions for different error-magnitudes
remained, in some cases the differences were larger still.
The models were also tested on the 5th order Chebyshev filter used in 1.3.1
and 1.3.2, with better worst-case results than with the Cauer filter, but still
failing to produce useable data.
In an attempt to find solutions for the cases that failed to converge, several
initial values of the optimization parameters were tested. This resulted in
improved matching of the transfer functions, but the problem with different
sets of solutions for different error magnitudes even larger.
Some attempts at increasing the order of the model further by extending the
model with reactive elements at the source or termination of the filter were
performed. This improved the convergence in some cases, but this was not
investigated further as the resulting filter was considered to be too different
from the original filter to be useful for analysis.
50
Studies on Tuning of Integrated Wave Active Filters
Chapter 7 – Tuning Strategies for Wave Active Filters
51
7 Tuning Strategies for Wave Active Filters
Unlike other active filter structures, wave active filters have two in-ports and
two out ports, with the desired transfer function realized between opposite
ports, and the complementary transfer function between adjacent ports. This
gives the opportunity to implement rather unique on-line tuning systems.
One possibility is to add a reference-signal to the input signal inside the stop
band of the filter, if possible at a zero in the transfer function, and use the
reflected signal for tuning. Some means of distinguishing this signal from the
signal being filtered out are required, which makes the “correlated tuning
loop” described in 2.3.1 well suited, as one can now add signal to the input in
order to ensure enough spectral contents for the loop to be stable. Another
alternative may be to use spread-spectrum techniques as described in 2.3.2
Another possibility is to add a reference-signal at the input port opposite to
the normal input port, with a frequency well within the passband of the filter,
preferably at a frequency where the transfer function has zero attenuation.
Since the filter has very little attenuation inside the passband, there will be
very little reflected signals at these frequencies, possibly making it easier to
distinguish the reference-signal from signals filtered out in the main filter
path. Using a reference-frequency inside the passband of the filter may also
improve the accuracy of the tuning, especially if more than one reference-frequency is used to accurately measure the shape of the pass-band.
If it is possible to create at least one wave two-port tunable to the range of all
the time-constants present in the filter, another possibility arises; one could
use a single reference two-port and a single gradient generation circuit in a
model-matching tuning scheme, as described in 3.2.5, tuning one two-port at
the time. As with normal model-matching tuning, assumes that there is
enough signal present in the filter when the tuning is performed, one should
probably detect the current level of input signal before attempting any tuning,
to avoid detuning the filter during periods of silence.
Most standard on-line tuning schemes would of course also work, at least
with the original buffer-based implementation, where the high gain-error sensitivity of wave-active filters isn’t a problem, as they are relatively insensitive
to other errors. When other implementations are used, an off-line tuning
scheme would probably be necessary to ensure that the gain of each two-port
is correct.
52
Studies on Tuning of Integrated Wave Active Filters
One straight-forward method would be just feeding a signal through the wave
two-port, for example at it’s -6dB frequency, where the same signal amplitude
should be output from both ports, when the filter is correctly tuned. Then the
gain can be adjusted until both levels are very close to the half the input signal
amplitude. However, designing a good peak-detector is non-trivial, and poor
performance in tuning-circuits is often the result of badly designed peakdetectors. With careful design and offset cancelling this simple method may
still be useable.
Another tuning strategy especially suitable for wave active filters is to reconnect parts of the circuit into resonant circuits, which are then tuned to a
desired center frequency. The gain of the two-ports can be adjusted until the
oscillation is stable in the linear range of the circuit, both ensuring accurate
gain tuning and reducing the problem of frequency-tuning-errors that usually
result when the amplitude of the oscillation is limited only by the nonlinearities of the circuit.
Chapter 8 – Conclusions and Future Work
53
8 Conclusions and Future Work
8.1 Tuning of Continuous-Time Integrated Filters
While many techniques for tuning continuous-time integrated filters have
been proposed, they all depend on the type of filter being tuned, the performance specification, and in many cases the environment in which the filter is
being used.
The more generic techniques for on-line tuning relay on using a reference-circuit for measuring parameters relevant to the performance of the filter being
tuned, rather than actually measuring on the actual filter itself. Due to the
mismatch between circuits on the same chip, the achievable accuracy is in the
order of 1%, strongly dependent on the filter being tuned.
On the other hand, if one wish to measure on the filter directly and no
assumptions can be made on the spectral contents of the input signal, two
possibilities remain. If the SNR required by the application is low, and special
reference-signals or implementations are used, it is in some cases possible to
have the reference-signal passing through the filter together with the input
signal, without excessive reduction of the SNR or severe interference from
the input signal on the tuning-circuit.
One alternative is to use an off-line tuning scheme, which assumes that the
filter can be taken out of the signal processing path occasionally for periodic
measurements and retuning (possibly with an identical filter used as stand in
while tuning is performed).
If the input signal can be assumed to have enough energy in the band used for
tuning, it is possible to measure the effects of the filter using this signal, and
use this information for tuning the filter. The tuning-circuit for performing
these measurements tend to become relatively complex, and in some cases
have a large area overhead due to the need for an additional reference filter.
8.2 Wave Active Filters
While Wave Active Filters in their most basic implementation do possess
good tolerance towards component variations, the implementation of the
time-constant control necessary to make them useable in integrated circuits is
not a trivial task. The attempts made in this thesis to find a more easily con-
54
Studies on Tuning of Integrated Wave Active Filters
trolled implementation, without significantly increasing the sensitivity have
not been successful. The studied structures may however still be useable if
adequate tuning-circuitry is used, but it is questionable if they represent an
improvement over other structures such as leapfrog-filters.
Even though WAFs were first proposed almost 30 years ago, very little work
has been performed implementing them. Furthermore, in these cases the
focus has mainly been on finding structures reducing the required circuit area,
with severely degraded performance as a result. A study of on-chip implementations using the original implementation using only buffers as active elements may be interesting, especially in combination with one of the tuning
methods suggested in chapter 7.
Chapter 9 – References
55
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