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Studies on Tuning of IntegratedWave Active Filters

Johan Borg

LiTH-ISY-EX-3401-2003

Linköping 2003-06-05

Studies on Tuning of IntegratedWave Active Filters

Master’s Thesisperformed at

Electronics SystemsLinköpings 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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ammanfattningbstract

The first part of this thesis contains a literature study of current tuning tech-niques for continuous-time integrated filters. These tuning methods are charac-terised by which quantity they measure, their dependence on certain character-istics of the input signal, or matching of components on chip. The structure ofthe different tuning schemes are explained. The merits and drawbacks as wellas achieved accuracies of previous works are summarised.

The second part is a study of wave active filters (WAFs), a less commonstructure for implementing active filters. In this structure the filter is realisedby simulating the forward and reflected voltage waves present in the prototypefilter. The main advantage of this is that the inherent low sensitivity of doublyterminated ladder-filters is better preserved than in many other structures. TwoMosfet-C realisations of Wave Active Filters have been suggested and high-level simulations have been used to compare them to the originally proposedimplementation as well as a leapfrog implementation.

Electronics Systems,Dept. of Electrical Engineering581 83 Linkoping

2003-06-05

—

LITH-ISY-EX-3401-2003

—

ttp://www.ep.liu.se/exjobb/isy/2003/3401/

Studies on Tuning of Integrated Wave Active Filters

Studie av avst¨amning av integrerade aktiva v˚agfilter

Johan Borg

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yckelordeywords tuning, integrated filter, wave active filter, WAF

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Abstract

The first part of this thesis contains a literature study of current tuning teniques for continuous-time integrated filters. These tuning methods are cacterised by which quantity they measure, their dependence on cecharacteristics of the input signal, or matching of components on chip.structure of the different tuning schemes are explained. The merits and dbacks as well as achieved accuracies of previous works are summarised

The second part is a study of wave active filters (WAFs), a less commstructure for implementing active filters. In this structure the filter is realisby simulating the forward and reflected voltage waves present in the prtype filter. The main advantage of this is that the inherent low sensitivitydoubly terminated ladder-filters is better preserved than in many other sttures. Two Mosfet-C realisations of Wave Active Filters have been suggeand high-level simulations have been used to compare them to the originproposed 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

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1 Introduction

1.1 Background

Even though continuous-time integrated filters are usually replacedswitched capacitor filters where feasible, many important applicatiremain, such as anti-aliasing filters for high-speed data-converters andchannel equalizers for hard-disk drives.

The main reason for using continuous-time filters is their speed. A compable switched capacitor filter for signals in the MHz-range or higher wourequire excessively high clock frequencies, with high power consumptionclock-feedthrough as a result. Furthermore, high-performance operatioamplifiers (OP-Amps) will be required to obtain settling-times sufficienlow for the switched capacitor circuits to reach steady state within haclock period.

On the other hand, the main reason for using switched capacitor filters isstability. Since all passive elements are realised using capacitors only, thequency characteristics will only depend on the capacitor sizes and theirtive accuracy, which are typically less than 0.1% [1], and the clock frequen

For continuous-time filters this is not true, both capacitors, and either retors or transconductors are used to realise the filter, the ratio of their sizesdetermine the overall frequency characteristics. Unfortunately, chip to cvariations 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 frequecontrol, “tuning”, to ensure that the filter meets the specification.

Integrated filter design is further complicated by the fact that high perforance filters are sensitive to component variations. Because of this, it is onecessary to introduce some type of control over other parameters in thter, in order to compensate for effects such as parasitic loads and devicematch.

The sensitivity to component variations is also highly dependent on the stture of the filter, for example lattice, filters are generally only suitablecrystal filters, as they are extremely dependent on element stability, wdoubly terminated LC-ladder filters are relatively insensitive to small chanin component values.

2 Studies on Tuning of Integrated Wave Active Filters

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1.2 Outline of this Thesis

• Chapter 2 - On-Line TuningChapter 3 - Off-Line TuningThese two chapters contain the results from a literature study on the ject of tuning of continuous-time filters.

• Chapter 4 - Wave Active FiltersA background on wave active filters as well as an initial study of theirperformance in respect of component variations.

• Chapter 5 - Mosfet-C Implementation of WAFsAttempts at finding a Mosfet-C implementation and the performance the resulting candidates.

• Chapter 6 - Mapping of S-parameter Errors to Passive ComponentsFurther studies of the relation between filter defects in S and compondomains.

• Chapter 7 - Tuning Strategies for Wave Active FiltersSome 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-timintegrated filters.

• Study possible MOSFET-C implementations of wave active filters.

• Investigate if it is possible map scattering-parameter errors of wave acfilters to component errors.

Chapter 2 –On-Line Tuning 3

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2 On-Line Tuning

Because the parameters of integrated active filters depend on tempersupply-voltage and ageing, a tuning method that is active at all times (refeto as “on-line”), is usually required. The opposite is off-line tuning, where tfilter is only tuned when inactive.

The most common way to tune an integrated filter is by using a “masslave” tuning scheme. One or more filters are used as reference, continutuned by a control circuit to meet some reference performance. The cosignals from this process can then be used for tuning the filter(s) procesthe 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 wilimited by the matching of the master and slave filters. Another problem whaving the reference filter and the tuning-circuit operating continuously ispossibility of undesired signals from the tuning process leaking into the msignal path.

2.1 Master-slave Frequency Control

2.1.1 Gm or R - only Tuning

Where the required frequency accuracy is low, simply making sure thattransconductances (for Gm-C filters) or resistances (for (R-)MOSFET-C) arcorrect provides a simple solution. Since variations in capacitor values aretaken into account, the accuracy of the filter after tuning will be limited by tprocess variations of the capacitor values, which is usually about 10% [2

Figure 1: The principle of master-slave tuning

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For example, in Fig. 2, the voltage difference Vb will make the transconduc-tor output a current, Iout=GmVb. At the same time, there is a voltage differencVb over the off-chip resistor Rext, resulting in a current I=Vb/Rext. If these cur-rents are equal, no current is going into the integrator. An incorrect transcductance Gm will cause a difference in currents, this difference will bintegrated 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 wbuilt. Since it is designed for hard-disk read-channel equalizing, no datacut-off frequency accuracy is available, as this is secondary to the group dripple.

Similarly, in [4] an elaborate scheme for tuning ratios of conductancestime constants are presented. Maintaining these ratios is in this case nsary to ensure that the filter meets the group delay ripple specification. Onother 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 switcapacitor equivalent, and thereby control a Gm/C or R/C ratio directly. In the-ory this would look like Fig. 3, but that approach is usually not realistic, dto the high clock frequency required if Gm and C have similar value to thoseused in the slave filter (which is preferable to achieve good matching). Tcan be solved by using the circuit in Fig. 4, which scales down the clockquency a factor N, by using two currents of a ratio 1:N.

Figure 2: Gm - only tuning

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A lowpass filter is usually required to reduce reference-signal leakage intoslave filter. While simple, these filters can become quite large, due to the lcapacitances required for large time-constants. In some cases an offcapacitor has been used [5].

Another option is to lock the time-constant directly to a reference clock, lin Fig. 5, where the capacitor C is charged with a current determined bytransconductor, and the peak reached during 1/2 clock cycle is comparethe voltage Vb [6].

Figure 3: Gm/C tuning by using SC-circuit

Figure 4: Improved Gm/C tuning using SC-circuit

Figure 5: Gm/C tuning by locking the time constant to the period of a reference clock

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For active R-C and similar filters, the circuit in Fig. 6 could be used, eithwith Vf directly controlling a bias to the mosfet-resistances in case of a Mfet-C filter, or through a comparator controlling a counter, in turn switchidifferent R or C elements in or out [8].

Here the current Vb/R through the resistance will be balanced against the crent -fVbCm transferred by the switched capacitor.

All methods discussed so far have the advantage of being very simple, anopposed to most other methods, the reference-signal is not required tosine wave with low distortion.

Using a reference clock of a frequency considerably lower than the operafrequency of the filter, like in Fig. 4, will reduce the problem with referencsignal leaking into the main signal path.

Accuracy will largely depend on offsets in the active components, but alsoachieving good matching with the slave filter. This may be difficult since tstructure of the master filter is fundamentally different from the slave filtThis may result in parasitics affecting the nodes differently, with a systemerror as result. Tracking of production spread and temperature variationsalso likely to be relatively low when these methods are used.

In [7] a 4th-order 10.7MHz bandpass filter was tuned to a frequency accuof 1% by using a circuit similar to that in Fig. 4.

In [8] a 14th-order Chebyshev bandpass filter operating in the 165-505range, was tuned to an accuracy of 1% by a circuit similar to that in FigHere a reference-frequency well above the operating frequency of the fiwas used.

In [9] three different 78kHz active-RC filters with 5 bit binary weigtheswitchable capacitor arrays, controlled by a circuit similar to that of Figoperating in a dual-slope mode were implemented. Frequency accuraci5% were obtained.

Figure 6: R-C filter version of tuning using SC-circuit

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2.1.3 Integrator and First-Order Filter Based Tuning

An ideal Gm-C integrator will have the transfer function

(2.1)

Solving for |H(ω)|=1, we get:

(2.2)

Which means that the unity gain frequency of the integrator will be

(2.3)

As described by Fig. 7, this can be used to control the Gm/C ratio, comparingthe peak level of the reference-signal before and after it has passed throreference integrator. If the Gm/C ratio is correct, the output from the integrator should have the same amplitude as the input. Any difference in amplituwill be integrated over time by the second integrator, and the control signaf

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 fi

For a non-ideal integrator it can be shown [10] that the frequency error wilbelow 0.1% if the DC-gain is larger than 40dB and the phase-error at ugain is smaller than 1 degree.

Figure 7: Tuning using unity-gain frequency of the integrator

H s( )Gm

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When this method is used, the input offset of the transconductor must beenough to keep it from saturating, since no DC loading or DC feedback eOne way of avoiding this problem has been proposed in [11]. In Fig. 8,new transconductor will simulate a resistance R=1/Gm, and the transfer func-tion becomes:

(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-filtering the result, for measuring a signal amplitude, seems arbitrary in mcases, but here the latter might have an advantage. This is because takinsquare of a signal with a relative amplitude of -3dB will result in an outpDC-level of half that of a 0dB input signal. A peak-detector is on the othhand designed to preserve a linear relationship between input amplitudethe output voltage, and will thereby produce an output of times thaa 0dB signal. In this case, when a ratio of the signals should be 3dB, immenting the attenuator after the squaring amplitude detector may impaccuracy, since it is usually easier to implement accurate integer ratios.

This type of tuning has also been implemented in [12], [13] and [14], for tuing different circuits, but no useful experimental data is available on tunperformance.

2.1.4 Phase-Locked Filter

The main feature of this method is that good matching between masterslave is relatively easy to obtain, since both are filters and can be built usimilar structures.

Figure 8: Tuning using a degenerated integrator.

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In Fig. 9, a sine-wave reference-signal is used as input to the master fiThe phase of the output signal from the filter is compared with that of theerence-signal. The phase comparison is carried out by multiplying thenals, as the DC-component of the product of two signals with the safrequency will depend on the phase difference. If there is a 90 degree pdifference the output will be zero. The output from the multiplier is integratover time, and used as the frequency control signal. This will effectively lothe phase-shift through the filter at 90-degrees, as a different phase shifproduce a DC output, which will be integrated until the control signal hchanged enough to correct the phase shift.

A second order lowpass filter is usually used for the master filter, as ithave a 90 degree phase shift at its -3dB frequency. This is true even wheslave filter is of a different type or order, because locking to a 90 degreeference usually simplifies design. Filters of higher order may also have mthan one frequency where the phase difference is 90-degrees. Thus, thepossibility that the tuning-circuit may converge to the wrong frequency (pvided the tuning range is sufficiently large).

Other types of filters may be used, however using a filter with a degphase shift at the reference-frequency usually simplifies the design. If180 degree phase shift is used, either the quadrature component of theence-signal or an additional 90 degree phase-shift will be required.

In some applications it might still be advantageous to use a notch-fiinstead [15], especially if the location of a zero in the transfer functionimportant. When using a notch-filter as a reference, the output sigapproaches zero as the frequency of the zero in the notch-filter approachefrequency of the reference-signal. This will theoretically reduce the reence-signal leakage to the main signal path and reduce the size of the Lter in the frequency control loop.

Figure 9: Tuning using a phase-locked filter

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The phase-comparator can be a major error source, as a phase-errordegree will cause a frequency-tuning-error of 0.5%, if the reference-filter2nd order lowpass with a Q-values of 2. Using higher Q-values will reduthis error, but may result in reduced matching of the master and slave filt

If the initial tuning error is large enough to make the reference-frequencywell inside the stop-band of the master filter, the amplitude of the input sigto the phase-detector will be low. If the phase-detector is based on directtiplication of the signals, the decreasing input signal amplitude will lead treduction of output signal amplitude. For large tuning errors, this effect wovertake the phase-detection and cause an overall decrease in output frophase-detector. Depending on the feedback loop design, this may causevergence problems. A solution for this problem is to decrease the Q-valuthe master filter, as this will make the slope of the phase shallower, andmake the variations in amplitude less dramatic. Alternatively, it shouldpossible to avoid this problem by using a feedback loop that contains angrator, as the sign of the phase signal will always be correct, even ifamplitude shows inconsistencies.

In [16] an integratorless feedback loop was used with this type of phadetector. This resulted in a requirement of Q<2 to ensure convergence o30% range.

In [17] a 5th-order elliptic 1.92MHz lowpass MOSFET-C filter was tuned,data on absolute frequency accuracy were presented, but the tempercoefficient of the cut-off frequency is said to be 100ppm/ C.

In [18] an 2nd order 78kHz lowpass active-RC filter using digitally programable current attenuators was tuned to an accuracy of 5%, of which the qtization error may account for 1-3%.

In [19] an unusually large ratio of master/slave cut-off frequencies was usthis resulted in relatively large temperature and supply voltage dependenfor the center frequency and Q-value.

2.1.5 Phase-Locked Oscillators

To eliminate the requirement of a low-distortion sine wave reference-sigand the absolute accuracy of the phase-detector, phase-locking of an ostor implemented with a structure similar to that of the slave filter, is oftused. However, in order to make sure the circuit forms a stable oscillator wthe active elements operating in their linear regions, new elements like nlinear negative resistances, modified transconductors or limiters are us

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required. These changes make a good matching to the slave filter hardachieve compared to a phase-locked filter. Another approach is to try to ka filter section oscillating by increasing the Q-value to infinity. This, howevalso tend to cause a systematic frequency error.

In Fig. 10 an oscillator is formed by inserting a limiter in the feedback lofrom the output to the input of a bandpass filter, which must have a passbgain larger than unity. The limiter will crop the peaks of the signal to solevel. This ensures that the amplitude of the input signal is low enough forfilter to be sufficiently linear. Too high input signal amplitude will make nolinearities in the filter significant, with a change of oscillation frequency aresult.

When the tuning is complete, the oscillator is phase-locked to the referesignal and any frequency error will make the phase error increase over tThis in turn will change the DC-output from the phase-detector and adjustcontrol signals for the filter. Because the phase error is the frequency eintegrated over time, no stationary frequency error will remain.

Depending on the phase-detector used, locking range may be limited toone octave, which is sufficiently wide to handle the tuning range of mostters. However, in some cases when this method is used with very wide-btuneable MOSFET-C filters, means for avoiding locking to harmonics mayrequired.

In [20] a 5th-order 3.4kHz elliptic lowpass filter was implemented, withproduction spread after tuning of 0.5%, and a temperature dependenc0.1% over the range 0-85 C.

In [21] a 4th-order 70MHz bandpass filter was implemented, with a systeatic frequency error of 1.5% and a production-spread of 1%.

In [22] a 1MHz 2nd order active-RC using programmable capacitor arrawith frequency errors within 2% after tuning were implemented.

Figure 10: Tuning using a phase-locked voltage controlled oscillator

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2.2 Master-Slave Q-value Control

For a pole Pk, the Q-value is defined as

(2.5)

For biquad filters, this is directly applicable to each biquad individually,they implement one pair of complex conjugated poles each. When a filterhigher order than two is implemented in a single structure, Q-values willdefined only for the realized poles, with no direct connection to the filimplementation as such.

In any case, making sure that the poles of a filter doesn’t move too far frtheir desired positions will be critical for ensuring that the shape of the paband remains acceptable.

The Q-values present in active filters are usually determined by a rbetween values of similar components, like Gm1/Gm2or C1/C2. Since the sizeratio between components of the same type is relatively insensitive to proand temperature variations, Q-values should also be relatively insensitivetherefore not require any tuning. This is usually true for low frequencies afor low Q-values, where the component ratios are small and the active cponents are nearly ideal. At higher frequencies and larger component ranonidealities, parasitics and process variations may cause considerable dtions from the desired Q-value.

The common methods of adjusting Q-values in a filter, are either adjusthe ratio of the component values that determines the Q-value, introducicontrollable (positive or negative) resistance element in the circuit, or in cof 2-stage active elements, adjusting a compensation circuit inside thement.

When the frequency and Q-value tuning are not entirely independent, thcontrol loop is usually made an order of magnitude slower than the frequecontrol to make sure that the Q-value tuning is preformed at the correctquency.

2.2.1 Phase-Locking an Integrator

It can be shown that a phase error in the active components of a filterhave considerable effect on the Q-value. When a single integrator is use

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master for frequency control, see section 2.1.3, this phase-error will causphase difference over the integrator (after frequency-tuning) to differ fromideal 90-degrees. This has been used in the tuning scheme presentFig. 11.

Here the reference signal and the output signals are converted to logic leand used as inputs to an xor-gate. If the phase difference is not 90-degthe output from the xor-gate will not have a 50% duty-cycle. This will causnon-zero average output current from the transconductor, charging orcharging the capacitor CI and thereby adjusting the control voltage VQ.

The accuracy of this method will depend on the achievable phase accurathe phase-detector. It should be remembered that only phase errors causnonidealities in the transconductor are measured and corrected, errorsnating from inaccurate component ratios, due to process variations or paics, are not.

In [10] a 4MHz 6th-order elliptic lowpass filter was tuned by this methothey claim good theoretical accuracy for the phase-detector based on corators and a xor-gate, but no experimental data on the performance of thvalue tuning is presented.

2.2.2 Amplitude Locking Passband Gain

Fig. 12 shows the most common way of implementing Q-value tuning, sply using that the passband gain of a 2nd order bandpass filter will be protional to the Q-value. If we assume that the mid-band gain is equal to thevalue, a too low Q-value will produce an output lower than that of the amfied reference-signal, this difference will be integrated over time, untilcontrol-signal VQ has changed enough to correct the Q-value. This signaalso used to control the slave filters.

Figure 11: Q value tuning using phase difference

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If a biquad filter is used as master in the phase-locked filter frequency-tunloop, a bandpass-filtered signal is usually already available in the circuit,erwise, a separate Q-value tuning master is used.

Q-value tuning is often used to compensate not only for nonidealities ofactive elements, but also for component mismatches caused by parasiticprocess variations. There have been implementations with one Q-maidentical to each stage in a chain of biquads. In [23] four stages were usemake sure that all stages were compensated correctly, instead of tryinscale the compensation circuits.

If a frequency-tuning-error is present, the reference-frequency will notexactly in the center of the passband. Because of this, the gain measwhen the reference-signal is feed through the filter will not be the passbgain of the filter. This will result in a Q-value tuning error, since the tunincircuit will make the meassured gain equal to the desired passband-gainthe actual passband gain will be forced to some different level. This errorbe approximately proportional to the Q-value, as the passband width isinversely proportional to the Q-value.

It has been suggested [24] that this error can be reduced (ideally eliminaby using the circuit in Fig. 13.

Figure 12: Passband amplification based Q-tuning

Figure 13: Improved passband amplification based tuning

V ref

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Here the change in VQ is calculated as

(2.6)

Where Vref and Vbp are the reference-signal before and after it has pasthrough the (bandpass) master filter.µ is the integrator gain.

When the tuning is complete, and no frequency error is present, both amtude and phase will be equal. In case a frequency-tuning-error is presthere will also be a phase shiftφ trough the filter, which will make this circuitadjust VQ until the following condition is meet:

(2.7)

This means that when the tuning is complete, the gain of the filter will be

(2.8)

However, for a second order bandpass filter

(2.9)

the phase shift trough the filter will be

(2.10)

Eq (2.9) and (2.10) gives

(2.11)

and the magnitude response as a function of the phase shift will be

(2.12)

Comparing this with Eq. (2.8), we now see that the filter will ideally be tunto the correct Q-value, even if the reference-frequency is not in the ecenter of the passband.

This method can actually be seen as an Least Mean Square (LMS) adapalgorithm implementation, where the output from the master filter is usedan approximative gradient signal.

VQ µ Vref Vbp–( )Vbp=

Vbp Vref φcos=

H φ( ) Q φcos=

H s( )ω0s

s2 ω0

Q------s ω0

2+ +

---------------------------------=

φtan Qω0

2 ω2+

ωω0-------------------=

H φ( ) iQφ i+tan

------------------- iQ φ φ i φsin+cos( )cos= =

H φ( ) Q φcos=


asDis-re-deing-

ire-cilla-hed.

ed,

f thefil-in

lter

h to

In [24] a 10.7MHz single biquad bandpass filter with a Q-value of 20 wmanufactured and a Q-value error of 0.75% was measured (after tuning).crete tests of a similar circuit with a Q-value of 10 indicated that a 3% fquency error would result in an 1.1% Q-value error. If a normal amplitucomparing Q-value tuning-circuit had been used, a 3% frequency-tunerror 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 requment of a separate Q-tuning master filter when using a phase-locked ostor for frequency control. According to [25] the method reduces tsensitivity to offsets in the tuning-circuit compared to the previous metho

A 2nd order 100MHz bandpass filter with a Q-value of 20 was manufacturand 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 ooscillations will be equal to the step-response from a first order low-passter, with a -3dB frequency of half the 2nd order filters bandwidth, as shownFig. 15.

In [26] it was proposed that this may be used for tuning the Q-value of a fias shown in Fig. 16

Here Vref is a square wave reference-signal, with a frequency low enougallow the filters to reach steady state after each transition.

Figure 14: Combined frequency and Q-tuning scheme

LPfilter

1/Qmasterfilter

+

-

+

V ref

LPfilter

f ctrl

VQ


tors,out-inte-

hean be

as-

The output signals from the two filters pass trough the envelope detecwhich produce an output proportional to the square of their inputs. Theputs from the envelope detectors are then compared and the difference,grated 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 tfilters have reached steady state, the signal leakage to the slave filters creduced.

In [26] a board level test circuit was built, and Q-tuning errors of 3-7% mesured.

Figure 15: 2nd order vs 1st order lowpass filter

Figure 16: Envelope based Q-tuning

0 2 4 6 8−1

−0.5

0

0.5

1

t

V(t

)

V ref

S/H &LP-filter

reference

(1st order)

filter

masterfilter

(2nd order)

envelopedetector

envelopedetector

VQ


thhip

ngensi-

.2, itop-

naln.

likenal

tog the

ofli-

theient

nver-

utput

es,

In [7] a 10.7MHz 4th-order biquadratic bandpass filter with Q=20 for bobiquads was implemented, with a systematic Q-tuning error of 20% and cto chip variations of 10%. This is attributed to offsets in the comparitransconductor, inaccuracies in the envelope-detection and frequency-stivity proportional to Q.

While not as accurate as the improved amplitude locking described in 2.2may well be comparable to the classic amplitude locking method and, if prerly implemented, provide an acceptable level of accuracy.

The low frequency of the reference-signal will help reduce reference-sigfeedthrough to the slave filter, and possibly reduce the power consumptio

2.3 True On-Line Tuning

Ideally, one would want to measure the characteristics of the actual filter,in off-line tuning, and at the same time be able to have both tuning and sigprocessing 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 similarthe methods described in 2.1.4 and 2.2.2 as it tunes the filter by observintransfer function of the filter at a single reference-frequency. Insteadactively providing the filter with a known input signal, and measuring ampfication and phase shift at the output, these parameters are derived frominput signal. This tuning method assumes that the input signal has sufficspectral contents at the reference-frequency, if this is not the case, cogence of the tuning loop can not be guaranteed.

For a linear system, the relation between the spectra at the input and ocan be written

(2.13)

where Gxy and Gxx are the cross power and input power spectral densitirespectively.

Gxy ω( ) Gxx ω( )H ω( )=


h

xx,

shiftsig-not

It can be shown [27] that signals Va,Vb, Vc and Vd calculated as

(2.14)

(2.15)

(2.16)

(2.17)

are orthogonal representations of the energy at the input (Vc and Vd) and out-put (Va and Vb). The signals will be low frequency or DC, with a bandwidtdetermined by the lowpass filter hLP. It can also be shown that

(2.18)

(2.19)

(2.20)

will be estimates of the average real and imaginary parts of Gxy and the Grespectively. The center frequency can then be locked by using either Vx orVy, depending on the filter tuned, as a measure of the error in phasetrough the filter, and integrate this signal to create the frequency controlnal. Amplitude, and thus Q-value, can similarly be created from the signalused for frequency-tuning, combined with Vref.

The proposed tuning system is shown in Fig. 17, where Vf (=Vx) is used forcontrolling center frequency of the filter, while VQ (=Vy-Vref) is used to forcethe gain to unity, in this case corresponding to a Q-value of 10.

Va y u( ) ω0u( )hLP t u–( )sin ud

∞–

t

∫=

Vb y u( ) ω0u( )hLP t u–( )cos ud

∞–

t

∫=

Vc x u( ) ω0u( )hLP t u–( )sin ud

∞–

t

∫=

Vd x u( ) ω0u( )hLP t u–( )cos ud

∞–

t

∫=

Vx t( ) Vb t( )Vc t( ) Vd t( )Va t( )–=

Vy t( ) Va t( )Vc t( ) Vb t( )Vd t( )+=

Vref t( ) Vc2

t( ) Vd2

t( )+=


tem

nd-

dB

t theay

d thedom

ncynal.er-

realThe

Here the signals Fox and Foy are the reference-signals, with a phase shift of 0and 90 respectively, x and y are the input and output signals of the syswhile 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 bapass filter with a Q of 10, implemented in a 2µm CMOS process. With a fullswing input signal a frequency-tuning-error of 0.2% and a gain error of 1.1was obtained.

2.3.2 Orthogonal Reference Tuning

If assumptions about the input signal, as in 2.3.1, can not be made, burequired signal to noise ratio is low, the tuning method proposed in [29], mbe an option for true on-line tuning of the filter.

Here an approximative orthogonality is created between the reference aninput signal by phase modulating a reference-signal with a pseudo ransequence before adding it to the input signal, as shown in Fig. 18.

This will result in the reference-signal being spread out over a frequeband, with the width determined by the rate of the phase modulation sigThe output signal from the filter is then multiplied by the modulated refence-signal and its quadrature components, producing estimates of theand imaginary parts of the transfer function at the reference-frequency.result is then used to tune the filter, as described in 2.3.1.

Figure 17: Tuning by using correlation of input and output signal

LPFilter

LPFilter

F (t)ox F (t)oy

LPFilter

LPFilter

F (t)ox F (t)oy

Vf VQ

2nd OrderBandpassFilter

x(t) y(t)

aVVbcVVd

aV

Vb

cV

Vd

aV cV

cV

Vd

Vd

Vb

-

-

Vf

VQ

°°


fil-ingnd-t aas

verstestref-the

cir-orsin-lly

A board level test circuit was built, tuning a 2nd order 10.7MHz bandpasster with Q-value of 100. To make the theoretical accuracy of the Q-tun10%, a carrier to reference (C/R) ratio of 20dB, using a control loop bawidth of 10-4 times the modulation rate was required. For the test circuimodulation rate of 10kHz and control loop bandwidths of 1.6Hz wselected.

A possible application for the proposed tuning scheme would be in receifor wideband-FM and QPSK (of low dimensions) modulated signals. Thecircuit was inserted in the signal path of a FM broadcast receiver, and theerence-signal was virtually undetectable during listening tests when onlymonaural part of the signal were used.

2.3.3 Tuning by Using Common Mode Signals

Integrated continuous-time filters are usually implemented as differentialcuits in order to improve linearity, by using differential transconductorsoperational amplifiers. If the filter was instead designed as two identicalgle-ended structures, with input and output signals feed differentia

Figure 18: Orthogonal reference tuning

x(t) TunableBandpassFilter

y(t)

ControlLoop Filter

LowpassFilter

ReferenceRecovery

Correlators

SpreadingSequenceGenerator

Vref

AutomaticGain

Control


t as a

-sig-me

w-e on

thenalsgnal

byandnaliffer-ig-

te

between them, one could in theory have a tuning reference-signal presencommon-mode signal in the filter [30].

Due to mismatch of the two single-ended filters, some residual referencenal will be present in the output signal, and the input signal will have soinfluence on the tuning-circuit.

In [30] a 7th-order equiripple filter using three biquads and a first order lopass filter was designed and simulated. The tuning-circuit only measurthe 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%,dynamic range would be 50dB if the levels of the input and reference-sigare equal, however, up to 80dB might be obtainable if the reference-silevel is reduced.

In [31] a single biquad 60MHz lowpass filter was tuned by this methodusing phase-locking for tuning the cut-off frequency as described in 2.1.4amplitude locking for Q-tuning as described in 2.2.2. The reference-sigwas added as common mode level at the input, and separated from the dential output by adding the outputs. For recovering the differential output snal a high CMR amplifier was used. For a 20mVp-p reference-signal added athe input a residual level of 300uVp-p was present at the output, this should bcompared to the input signal range of 2Vp-p

Chapter 3 –Off-Line Tuning 23

areleup,

ed,ger

heycy ofave,wer

filterfilterfil-

onswas

A)term

ionstly

3 Off-Line Tuning

As opposed to on-line tuning processes, like master-slave tuning, whichactive while the filter is operational, off-line tuning is only performed whithe filter is inactive, which may only be when the system is powereddepending on the application.

The advantage of off-line tuning is that the main filter is characterisinstead of a reference circuit, thus, the accuracy of the tuning will no lonbe dependent on the matching of these circuits

While the methods described herein are mostly suited for off-line tuning, tcan in theory be used in a master/slave circuit. However, as the accuramaster-slave tuning is limited by the matching of the master and the slusing these methods are probably hard to justify, due to their larger poconsumption 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 thewas used, where the center frequency of a 16th-order 450kHz bandpasswas tuned to an accuracy of 0.33%. A step was applied to the input of theter, and by using digital counters, the frequency of the resulting oscillatiwas measured and the control voltage adjusted accordingly. This processcarried out 3 times in a row, to reach the desired accuracy.

Since the chip was to be used in a time-division multiple-access (TDMenvironment, tuning could be repeated often enough to ensure that long-parameter variations would not be a problem.

When implementing this method, one should remember that the oscillatresulting from a step on the input will have a frequency deviating slighfrom the center frequency of the filter.


th-nedig-

is afilterearthe

rolsedare

igh-

lteron-of

out-

adors

fre-+1telytheith

3.1.2 Forced Oscillation

Another, not very successful method, was proposed in [33]. A 250MHz 8order biquad R-C filter was forced to oscillate by changing the gain of oamplifier in each biquad. The oscillation-frequency was measured usingital counters.

This resulted in a frequency-dependent systematic error of 5-10%. Thislarger shift than can be accounted for by the change of Q-value when thewas forced to oscillate. One possible explanation might be the nonlineffects encountered when the oscillation is limited by the linear range offilter.

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 continput of the filter was proposed. A constant frequency reference-signal is uas the input, and the resulting amplitude-variations of the output signalobserved and used to tune the filter. This method is only applicable for hQ filters, with a well-defined peak in the amplitude-response.

This method is implemented by sampling the control voltage when the fiamplification passes a level slightly below the peak level, once for rising ctrol voltage and once for falling control voltage, and using the averagethese voltages for controlling the filter. It is also possible to use the peakput 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 biqubandpass filter tuneable over 105-120MHz, with frequency-tuning-errbelow 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 aquency-divider controlled by the tuning-circuit, producing 2 frequencies Nand N-1 times the reference-frequency was proposed. N/2 is approximaequal to the desired Q-value, and N times the reference-frequency isdesired center frequency of the filter to be tuned. A second order filter wbandpass and lowpass outputs are assumed.


ondase-e to:

hat ffor

indi-

hreeoris

As seen in Fig. 19, the 3dB frequencies of the bandpass filter will correspto phase shifts of -45 and -135 degrees in the lowpass filter. These phshifts are measured, and the frequency-tuning loop is designed to converg

(3.1)

and the Q-tuning loop should converge to:

(3.2)

It should be remembered that the statements that Q is equal to N/2, and t0

equal to N times the reference-frequency are only approximately true,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 thevidual 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 tfrequencies (N-1,N,N+1)ωref are used, with the signal attenuated by a factof two whenΝωref is being generated. The center frequency of the filtertuned to make the output amplitude from(Ν−1)ωref and(Ν+1)ωref equal, andthe Q-value is tuned to make the amplitude fromΝωref equal to that of one ofthe other reference-signals, locking the -6dB bandwidth to 2ωref.

Figure 19: Phase-frequency relation of a 2nd order bandpass filter

ω

ω

|H ( )|ωBP

arg(H ( ))ωLP

3dB

0

-135

-90-45

-180

φ N 1+( )ωr( ) φ N 1–( )ωr( )+ 180°–=

φ N 1+( )ωr( ) φ N 1–( )ωr( )– 90°–=


wasr of

Q-uits

ndrts of

d totherply

at aas

of the

neand

fre-tun-asrrent

erig-ula-

nly.

s in

A second order 200MHz bandpass filter with a desired Q-value of 28.6manufactured, and a frequency-tuning-error of 0.25% and a Q tuning erro3% was measured.

3.2.4 Isolation of Sub-Circuits

If the shape of the passband is important, tuning center frequency andvalue of the filter may not be enough. One approach is to isolate sub-circin the filter and tune them individually.

In [39] tuning of a leapfrog filter by isolating resonant loops in the filters, aseparately measuring their resonance frequencies was proposed. The pathe filter that are not part of a resonant loop may either be reconnecteform one, or they may be tuned by the methods described in 2.1.3. Anoapproach is to isolate the filter completely into first order sections, and apthe method from 2.1.3 to each part individually.

In [40] a 6th-order narrow band Chebyshev filter was tuned one resonatortime, by shunting the others to ground. A frequency-tuning-error of 3% wmeasured, which is suggested to be caused by nonideal characteristicsphase-detector used.

In [41] a 4th-order, 21.4MHz butterworth filter was tuned by isolating oresonator at a time, and employing the tuning schemes described in 2.1.42.2.2 for frequency and Q-tuning, respectively. They obtained a centerquency accuracy of 0.014%. Here a mixed-signal implementation of theing-circuit was used, where a D-type flip-flop replaced the multiplierphase-detector, and a successive approximation scheme controlling cuDACs replaced the integrator.

3.2.5 Model Matching

In [42] the use of a model-matching algorithm for tuning continuous-timintegrated filters is proposed. Model-matching algorithms in general are oinate from control theory, where they are used to create a model of a simtion model of a physical system, by observing input and output signals o

Ideally this type of method can tune the position of all the poles and zerothe filters.


effi-

lter,riva-

tooer,yen-

by a

d as

c

r cantate-

for

r isd-

In this case the least mean square (LMS) algorithm is used, where the cocients of the filter are updated by

(3.3)

where e(t) is the difference between actual and ideal output from the fiand is the gradient signal. The gradient signal is defined as the detive of the output signal with respect to parameter bn, thus, if both e(t) and

are positive at a given instant, the output signal at this instant islow, but if parameter bn had been larger, the output would have been highso increasing bn, as indicated by will reduce the error. Normallthe product of the gradient and the error signal is used, but in this implemtation is used in order to simplify the multiplication circuit.

Linear time-invariant systems, such as ideal filters, can be describedstate-space representation

(3.4)

where u(t) is the input signal, y(t) the output, and xi(t) the internal states ofthe filter.

Filters for generating gradient signals (gradient filters) can then be derive

(3.5)

The gradient for Aij can be found from the state xi(t) in a gradient filter withthe state xj(t) in the main filter as input, similarly, the gradient for bi is foundas the state xi in a gradient filter with u(t) as input signal. The gradients forand d are the states of the main filter and the input signal, respectively.

Depending on the filter structure used, the tunable parameters of the filtebe found from more or less simple relations to the parameters of the sspace description. In the article an orthonormal ladder filter was used,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 filterequired, which takes its input from different points in the main filter, depening on which gradient signal is being generated.

bn˙ t( ) 2µe t( )φbn

t( )=

φbnt( )

φbnt( )

e t( )φbnt( )

e t( )sign φ( bnt( ) )

sX s( ) AX s( ) bU s( )+=

Y s( ) cTX s( ) dU s( )+=

Agrad AT

= bgrad cT

=

cgrad bT

= dgrad d=


putal toAC,

Hereto

uts.ved

ed,

een

hilefil-thethe

ffi-er-ers.

The proposed tuning-circuit is shown in Fig. 20 (a), the predetermined inis generated by a pseudo random number generator followed by a digitanalog converter (DAC). The reference-signal is generated by another Dfrom a table of precalculated values.

This table of precalculated values is generated as shown by Fig. 20 (b).the desired continuous-time filter is simulated, and a digital filter tunedminimise the difference between the continuous and discrete filter outpWhen the tuning is complete, the output from the digital filter can be saand used for tuning the real continuous-time filter.

A working discrete tuning-circuit, tuning an integrated filter, was constructbut no performance measures other than time to tune the filer, are given.

Use of a dithered linear search algorithm for tuning filters has recently bproposed [43], eliminating the need for large gradient filters.

Adaptive tuning techniques can in theory also be used for tuning a filter wit is processing signals. This is done by implementing one more identicalter, which is first tuned by an other method. This second filter is then feedsame input as the main filter, and the adaptive algorithm is used to tunemain filter until the output signals are identical. If the input signal has sucient spectral contents, it would in theory be possible to tune the filter pfectly using this method, as it does not depend on the matching of the filt

Figure 20: Tuning by LMS model matching

IdealReference

Filter

TunableFilter

AdaptiveTuning

Algorithm

u(t) y (t)

e(t)

(t)δReference

SignalGenerator

TunableFilter

AdaptiveTuning

Algorithm

u(t) y(t)

e(t)

PredeterminedInput

+(Continuous)

(Digital)

PredetermiedInput n

+

(t)δn

(a) (b)

Chapter 4 –Wave Active Filters 29

anave

inbed

R

o be

as:

t of

tarybe

4 Wave Active Filters

4.1 Introduction to Wave Active Filters

Wave active filters (WAF) was first proposed in [44], in an attempt to findactive filter structure with the same insensitivity to coefficient errors as wdigital filters (WDF). Instead of simulating passive filter components, asgyrator-C filters, or node voltages as in leapfrog filters, the filter is describy the forward and reflected voltage waves.

Starting from the generic two-port N in Fig. 21 (with port resistancesi,i=1..2), incident waves A and reflected voltage waves B are defined as

(4.1)

Although different port resistances are possible, they will be assumed tequal in all cases discussed here.

The relationship between A and B is described by the scattering matrix S

(4.2)

The basic element when building wave active filters is the wave equivalena series inductor L, which can be shown to have the scattering matrix S:

(4.3)

with L=2Rτ, where R is the common port resistance. Τhis can be interpretedas a lowpass filter from input to reflected wave signal, and a complemenhigh pass filter for the transmitted wave signal. This functionality mayimplemented using the circuit in Fig. 22.

Figure 21: Generic two-port

NR1 R2 V2V1

A1

1B

I 1 I 2A2

B2

Ai Vi Ri I i+=

Bi Vi Ri I i–=

B1

B2

SA1

A2

=

S1

1 sτ+-------------- sτ 1

1 sτ=


itorsals.

ected

As shown in Fig. 23, series and parallel connected inductors and capaccan be created from this element by swapping outputs or inverting signMore complex elements like parallel series resonators and series connparallel resonators can also be realised from these blocks.

Figure 22: A simple implementation of the wave two-ports used in WAFs

Figure 23: Wave two-port equivalents of passive components

A1

1B

A2

B21 1

τ

τ

τ-1

τ

C= /2Rτ

C=2 /Rτ

L=R /2τ

-1

-1

-1

τ

L=2Rτ

C= /2Rτ

L=2Rτ1

2

L=R /2τ

C=2 /Rτ 1

2

τ2

1

τ

τ2

1

-1

-1

A2

B1

A2

B1

B2

A2

B2

A2

A2

B2

A2

B2

2B

A1

A1

2B

1B

A1

1B

A1

1B

A1

1B

A1


iva-gea.

n in

hasbeen

om-

teris-has

Terminating one port of the two-port adaptor with the resistance R is equlent to having no reflected signal from B to A at that port. The output voltawould be directly available on port B. Similarly, connecting one port tosource with impedance R, simply means feeding the signal directly into A

For example, the Chebyshev and Cauer filters, both of the 5th order, showFig. 24,can be realized as active wave filters according to Fig. 25

4.2 Sensitivity

The performance of wave active filter implementations presented so farbeen worse than expected [45], and it was suspected that this might havedue to lack of reciprocity in the wave two-ports, caused by unavoidable cponent variations.

Reciprocity basically means that a two-port has the same transfer-charactics in both directions, when the (possibly different) port impedancesbeen accounted for.

Figure 24: 5th order Chebyshev (a) and Cauer (b) lp-filters.

Figure 25: WAF realisation of 5th order Chebyshev (a) and Cauer (b) filters

R

R

L1 L3 L5

C2 C4VIN U VOUT

R

R

L1 L3 L5

C2VIN U VOUT

L2

C4

L4

(b)

(a)

τ -1L

V'OUT

VIN

OUT

INV'

V

1

τL3

τL5

τ

-1

-1

-1

-11

τ2

τ3

τ4

τ5

VOUTV'OUT

VIN V'IN

τL2

τC2

-1

τL4

τC4

-1

-1

(a)

(b)


proc-

-

inwith

ix S

setsof the2. Inencymed

t areom-

For a wave two-port where both ports have the same port resistance, reciity simply means that the transfer function from port A1 to port B2 is identicalto 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 shownFig. 25 (based on Chebyshev and Cauer lowpass filters), were simulateddifferent types of errors introduced.

4.2.1 Time Constant Errors

The time constant errors were created by replacing the scattering matrfrom Eq (4.3), describing the ideal wave two-port for an inductor, with:

(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, 10000of parameters were tested, and the largest and smallest absolute valuesamplitude responses were plotted for 1000 frequencies in the range 0 tothe graphs presented the curve in the middle represents the ideal frequresponse. These and all simulations in the following chapters were perforusing 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-pormaintained, as this ensures that reciprocity[46] is preserved for all the cponents derived as shown in Fig. 23.

In Fig. 28, the all errors in a two-port are equal.

S

sτe1

1 sτe1+-------------------- 1

1 sτe2+--------------------

11 sτe3+--------------------

sτe4

1 sτe4+--------------------

=


Figure 26: Frequency response for the Chebyshev and Cauer filters with independent time-constant errors

Figure 27: Frequency response for the Chebyshev and Cauer filters with reciprocity preserving time-constant errors.

0 1 20

0.2

0.4

0.6

0.8

1

Chebyshev

w

|H(w

)|

0 1 20

0.2

0.4

0.6

0.8

1

Cauer

w

|H(w

)|

0 1 20

0.2

0.4

0.6

0.8

1

Chebyshev

w

|H(w

)|

0 1 20

0.2

0.4

0.6

0.8

1

Cauer

w

|H(w

)|


isther ofnt touc-h fil-

n below

ilar

While there is a marginal positive effect of maintaining reciprocity, thchange is highly marginal and hardly visible from the plots alone. Fromvery marginal result of reducing the number of error-sources by a factotwo, one could suspect that there are other ratios that are more importamaintain. Some attempts at finding such ratios were made, with limited scess, as the results seemed to depend on which type of component, whicter structure and where in the filter the component was.

If all the added errors are kept equal in each wave two-port, any errors camapped directly to component errors in the LC-filters, with the expectedsensitivity as a result.

4.2.2 Gain Errors

In order to evaluate the effect of gain errors in the wave two-ports, simtests as in 4.2.1 were conducted, this time the scattering matrix

(4.5)

was used, where e1..4 are the error terms, en=1 when no error is present.

Figure 28: Frequency response for the Chebyshev and Cauer filters with equal time-constant errors

0 1 20

0.2

0.4

0.6

0.8

1

Chebyshev

w

|H(w

)|

0 1 20

0.2

0.4

0.6

0.8

1

Cauer

w|H

(w)|

S

sτe1

1 sτ+--------------

e2

1 sτ+--------------

e3

1 sτ+--------------

sτe4

1 sτ+--------------

=


.99 to000

t are

The same number of parameters was tested (10000) and error range (01.01) was used, with the largest and smallest absolute value plotted for 1frequencies.

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-pormaintained, in order to ensure that reciprocity is preserved.

Figure 29: Frequency response for the Chebyshev and Cauer filters with independent gain errors

Figure 30: Frequency response for the Chebyshev and Cauer filters with reciprocity preserving gain errors.

0 1 20

0.5

1

1.5Chebyshev

w

|H(w

)|

0 1 20

0.5

1

1.5Cauer

w

|H(w

)|

0 1 20

0.5

1

1.5Chebyshev

w

|H(w

)|

0 1 20

0.5

1

1.5Cauer

w

|H(w

)|


ve-nentthe

rorsnlyssi-ea

The conclusions from 4.2.1 applies here too, only a very marginal improment was seen from maintaining reciprocity, and the observed compodependency of which, if any, relations should be maintained to reduceeffect of errors, seemed to be roughly the same.

However, one should remember that the even distributions of random erused in this and previous section, combined with plots of min and max, ogive an idea about the worst case effects of errors, which may be highly pemistic if some errors interact strongly. However, it does give a rough idabout the relative importance of the different errors.

Chapter 5 –Mosfet-C Implementation of WAFs 37

s andty in

ionster 2

sist-.

s inanceby a

byion.d byst asOP-

ll

anytived in

tionunddif-

5 Mosfet-C Implementation of WAFs

5.1 Background

The chip to chip absolute variations of transconductances, capacitanceresistances are considerable. This results in a cut-off frequency uncertainthe order of 10-30%, if no correction is applied.

Fortunately, it is usually rather straight-forward to measure these variatand adjust the filter accordingly, using the techniques described in chapand 3, improving the final accuracy an order of magnitude, or more.

However, this does require that it is somehow possible to control either reances, conductances or capacitances, depending on the implementation

In the case of Gm-C filters this is straight-forward, as the time-constantthe filter will be determined by capacitances and the output transconductof the active elements. The transconductances are in turn determinedbias voltage which can easily be changed.

For active-RC filters the time-constant control is usually implementedrealizing resistances with mosfet transistors working in the triode regUnfortunately these resistances are not linear. This problem is be reduceonly implementing part of each resistance as mosfet transistor, and the rea passive resistance in series. By connecting the active part closest to theAmp input, the voltage amplitudes over it will be low, with improved overalinearity as a result.

Wave active filters implemented as described in the previous chapter lacksuch low-voltage node, making this type of control less useful. An alternacontrol-strategy is to have a bank of discrete valued components switcheor out to obtain the desired time-constant.

Based on this, it would be interesting to create a R-mosfet-C implementaof the wave two-port, where all resistors are connected to a virtual gronode at the input of an OP-Amps. In the study only structures suitable forferential implementations were considered.


wo-n in

uitshe

f onet is

5.2 Possible Structures

In an attempt to find a suitable structure, the signal flow graph of a wave tport was transformed in various ways, the resulting graphs are showFig. 31.

These were mapped to differential mosfet-C structures, the resulting circfor Fig. 31 C, A and F are shown in Fig. 32, 33 and 34, respectively. All tother 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 oOP-Amp are equal. The time-constant of the resulting wave two-porτ=RC.

Figure 31: Signal flow graphs for the basic wave two-port

1/(1+st)

A2

B1A1

B2

st/(1+st)

A2 B1

A1 B2

A

2

B1

A

1

B2

1/st

1/st

st/(1+st)

st/(1+st)

A

2

B

A

1 2

1/st

1/stst/(1+st)

st/(1+st)

B1

A

2

B

1A

1

B

2

1/(st+1)

st/(1+st)

A

2 B

1

A

1 B

2

1/(st+1)

st/(st+1)

A

2

B

1A

1

B

2

1/(st+1)

1/(1+st)

A

2 B

1

A

1 B

2

1/(st+1)

1/(st+1)

1/stA2 B1

A11/st

B2

1/st

A2 B1

A1

1/st

B2

1/st

A1

B 1

A2

1/st

B2

1/st

A1 B1

A2

1/st

B2

(A) (B)(C)

(D) (E) (F)

(G)(H) (I)

(J)(K)

(L)


Figure 32: Two OP-Amp realisation of the wave two-port (C)

Figure 33: Tree OP-Amp realisation of the wave two-port (A)

Figure 34: Four OP-Amp realisation of the wave two-port (F)

+A1

-A 1

+A2

-A 2

+B1

-B 1

+A1

-A 1

+A2

-A 2 +B2

-B 2

+A1

-A 1

+A1

+A2

-A 2

-A 1

+A2

-A 2

+B2

-B 2

+B1

-B 1

+B2

-B 2

+B1

-B 1

-A 1

+A1

+A2

-A 2


ruc-ionsypeRCple-

thetputmpingon-

fi-edticalpo-via-

ual,pref-to be

95thds to

r intionellle-

5.3 Sensitivity to Component Errors

In order to evaluate the sensitivity to component value errors for the sttures in section 5.2, monte-carlo analysis based on circuit level simulatwas performed on an 5th order Cauer filter. The filter used is the pi-tequivalent of the t-mode Cauer filter used earlier. Four different active-implementations were examined, three wave active and one leapfrog immentation (for comparison).

The first wave-active implementations used the circuit in Fig. 22, withunity gain buffers implemented as single-ended OP-Amps with the ouconnected to the inverting input. The second WAF used the two OP-Aactive-RC implementation of Fig. 32. The last WAF was implemented usthe three OP-Amp implementation of Fig. 33, where only a single time-cstant is used.

Finding realistic figures of component variations within a chip proved difcult; according to [1] a matching of 0.01% can be achieved for untrimmcapacitors on the same chip. However, this is the matching between idencomponents. No useful figures on accuracy of non-integer ratioed comnents were found, for the simulations a matching error with a standard detion 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 eqat the expense of capacitor ratios, but according to [1], this seems to beerable, as the achievable matching of equal sized resistors was claimedin the order of 0.1%.

The results are shown in Fig. 35. Dashed lines represents the 5th andpercentiles when only resistor errors are present, dotted lines corresponcapacitor errors, and solid lines the combined errors.

One should note that the buffer-based wave active filter is clearly superiothis respect. The 3 OP-Amp/two-port (single time-constant) implementahave similar sensitivity to capacitor variations, which makes it relatively wsuited for MOSFET-C implementation, if adequate tuning-circuitry is impmented to correct the resistance values.


in anthe

l-

as a

d to

5.4 Sensitivity to OP-Amp Bandwidth Variations

In the previous chapter ideal OP-Amps have been assumed, however,actual implementation the presence and location of poles and zeros intransfer function of the OP-Amp will influence the transfer function of the fiter.

In this chapter an OP-Amp based on example 5.7 in [47] is used, which htransfer function of

(5.1)

whereωz=120MHz, ωp1=4.2kHz,ωp2=143MHz and A0=80dB. This resultsin an unity gain frequency of about 100MHz. These values are later scale

Figure 35: Component error sensitivity of 5th order Cauer filters

0 1 20

0.5

1

1.5WAF, using buffers

w

|H(w

)|

0 1 20

0.5

1

1.5WAF, 2 OP−Amps/two−port

w

|H(w

)|

0 1 20

0.5

1


w

|H(w

)|

0 1 20

0.5

1

1.5leapfrog

w

|H(w

)|

H s( )A0 1 s

ωz------+

1 sωp1---------+

1 sωp2---------+

-------------------------------------------------=


nge

theP-ta-

50of

ts theency

sur-

areorenlyasitic

tothethen.

woe 3-nal

they.

er tostop-es the

obtain unity gain frequencies suitable for the normalized frequency raused herein.

During the initial tests with non-ideal OP-Amps it was determined thatMOSFET-C implementations require the unity-gain frequency of the OAmps to be about 2.5 times higher than for the unity gain buffer implemention.

Because of this, two different unity-gain frequencies are used (20 andtimes the cut-off frequency of the filter), in order to make the comparisonsensitivity to pole/zero location variations later more meaningful.

The results can be seen as part of Fig. 36, where the solid line represenideal frequency response, while the dashed line represents the frequresponse when the nonideal OP-Amps are used.

The large difference between ideal and actual frequency response is noprise, 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-Ampsknown when the filter is designed, it is possible to adjust the filter to restthe desired transfer function. Unfortunately, in real implementations oapproximate values are available, as pole/zero locations depend on parcapacitances and the loading of the OP-Amp.

The actual adjustment was performed by applying a minimax optimizationthe filters, where the error function to be minimized was calculated asmaximum of the largest absolute differences of the transfer functions inpassband, and the largest violation of the minimum stop-band attenuatio

In the buffer and 2-OP-Amp implementations of the wave-active filter the ttime constants of each wave two-port was changed independently. In thOP-Amp implementation only one time constant is available, so an additiogain parameter equal for both outputs was introduced.

For the leapfrog filter all the capacitors in the implementation (not onlytime constants from the original Cauer filter) were changed independentl

It was necessary to introduce these additional degrees of freedom in ordbe able to restore the passband shape. Still, only the attenuation of theband, not the exact shape, was preserved. One may note that in all caszeros have been moved away from the imaginary axis.


singsolided to

rors

oca-f thenor-de-

The result is shown in Fig. 36, dashed lines represents the original filter unonideal OP-Amps, dotted lines the desired frequency response, and thelines the response of the filter after component values have been adjustcompensate for the OP-Amps.

The adjusted filters showed marginally higher sensitivity to component erin tests performed in the same manner as in section 5.3.

In an attempt to study the effects of differences between the pole/zero ltions used when the filter was adjusted, and the real pole/zero locations oOP-Amps, the filters were simulated as described above, this time withmally distributed variations with a standard deviation of 10% added inpendently to the position of the poles and zero to all the OP-Amps.

Figure 36: Effects and compensation of nonideal OP-Amps

0 1 20

0.5

1

1.5leapfrog

w

|H(w

)|

0 1 20

0.5

1


w

|H(w

)|

0 1 20

0.5

1


w

|H(w

)|

0 1 20

0.5

1


w

|H(w

)|


imu-be acor-inal

esns,her onring

sateivesing

The results are shown in Fig. 37, where the 95th percentiles from 5000 slations are shown. One should remember that in this case, the result willdistribution around the frequency response of the OP-Amp bandwidth-rected filters, which have stop-bands with different shapes than the origfilter.

Considering the OP-Amps in the buffer implementation have a 2.5 timlower unity gain frequency than the OP-Amps in the other implementatiothis implementation performs very well, with no peaking at the end of tpassband, and only marginal cut-off-frequency change. The leapfrog filtethe other hand must also be considered relatively well behaved, considethe magnitude of the uncompensated error.

One should also remember that the cut-off frequency-tuning will compenfor the cut-off frequency inaccuracy, while filters suffering from excesspassband ripple usually implements a Q-value tuning-circuit for suppresthese changes.

Figure 37: Pole/zero position sensitivity of 5th order Cauer filters

0 1 20

0.5

1


w

|H(w

)|

0 1 20

0.5

1


w

|H(w

)|

0 1 20

0.5

1


w

|H(w

)|

0 1 20

0.5

1

1.5leapfrog

w

|H(w

)|

Chapter 6 –Mapping of S-parameter Errors to Passive Components 45

trixiving

andtwo-

d ined)

6 Mapping of S-parameter Errors to PassiveComponents

In order to better understand why a specific error in the scattering maaffects the transfer function the way it does, attempts where made at dera 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 gaintime-constant errors into the scattering matrix corresponding to the waveport description of a series inductor.

This proved to be non-trivial, for time constant errors of the same type usesection 4.2.1, recalculating the scattering matrix (with error terms includinto the admittance-matrix Y yields:

(6.1)

(6.2)

(6.3)

(6.4)

The gain errors from 4.2.2 result in:

Y111R----

2 1e1sτ

1 e4sτ+--------------------+

1 11 e2sτ+( ) 1 e3sτ+( )

--------------------------------------------------–e1sτ 2 2e1 e4+( )sτ+( )

1 e1sτ+( ) 1 e4sτ+( )--------------------------------------------------------+

------------------------------------------------------------------------------------------------------------------------- 1–

=

Y121R---- 1

1 e2sτ+( )------------------------- 2

1 11 e2sτ+( ) 1 e3sτ+( )

--------------------------------------------------–e1sτ 2 2e1 e4+( )sτ+( )

1 e1sτ+( ) 1 e4sτ+( )--------------------------------------------------------+

-------------------------------------------------------------------------------------------------------------------------=

Y211R---- 1

1 e3sτ+( )------------------------- 2

1 11 e2sτ+( ) 1 e3sτ+( )

--------------------------------------------------–e1sτ 2 2e1 e4+( )sτ+( )

1 e1sτ+( ) 1 e4sτ+( )--------------------------------------------------------+

-------------------------------------------------------------------------------------------------------------------------=

Y221R----

2 2 11 e1sτ+--------------------+

1 11 e2sτ+( ) 1 e3sτ+( )

--------------------------------------------------–e1sτ 2 2e1 e4+( )sτ+( )

1 e1sτ+( ) 1 e4sτ+( )--------------------------------------------------------+

------------------------------------------------------------------------------------------------------------------------- 1–

=


itsersur-

dingban-

ffer-hesel to

, inza-or anet-

of

anditialrigi-et torgere left

(6.5)

(6.6)

(6.7)

(6.8)

While it is theoretically possible to realise any reciprocal two-port fromimpedance or admittance matrix description, the complexity and high ordof the expressions will in this case result in complex networks. Since the ppose of this study was to better understand the mapping of errors, not finan exact equivalent, the attempts to find an analytical equivalent was adoned.

6.2 Approximate Mapping by Optimization

In an attempt to find an approximative passive component equivalent, dient types of networks based on doubly terminated LC filters were used. Tnetworks were created by adding more components to a LC-filter identicathe filter the WAF being modelled was first based on.

The transfer function of this new network was compared to that of a WAFwhich an error had been inserted in one of the two-ports. Minimax optimition was used to make the absolute value of transfer functions identical flarge number of frequencies, by adjusting the component values in thework based on a LC-filter.

A number of different structures for modelling the errors were tried, somewhich will be described here.

The simplest structure just included resistors in series with every inductorconductances added in parallel with each series resonant circuit. The invalues for the new components were selected to make it identical to the onal LC-filter, that is, both series resitances and parallel conductances szero. For the optimization the L and C values were constrained to be lathan or equal to zero, while resistances and conductances values werunbounded.

Y111R----

e2e3 e1 1–( )sτ 1–( )– 1 1 e4+( )sτ+( )e1 1+( )sτ 1+( ) 1 1 e4+( )sτ+( ) e2e3–

-----------------------------------------------------------------------------------------------=

Y121R----

2e2 1 sτ+( )e2e3 e1 1+( )sτ 1+( )– 1 1 e4+( )sτ+( )---------------------------------------------------------------------------------------------=

Y211R----

2e3 1 sτ+( )e2e3 e1 1+( )sτ 1+( )– 1 1 e4+( )sτ+( )---------------------------------------------------------------------------------------------=

Y221R----

e2e3 e1 1+( )sτ 1+( )– e4 1–( )sτ 1–( )e1 1+( )sτ 1+( ) 1 1 e4+( )sτ+( ) e2e3–

-----------------------------------------------------------------------------------------------=


r-heo 2.

filter

ses.ddi-

endsnsferter.

hen

n seeant if

ple-

g ofses

The error functionε minimized by the optimization was computed as

(6.9)

Where H0 is the desired transfer function, Hm is the transfer function of themodel filter andε0 is the initial error, used in consistent results from convegence limits for different magnitudes of errors in the wave active filter. Ttransfer functions was calculated for 1000 frequencies in the range of 0 t

The same Cauer filter as in 4.2.1 and 4.2.2 was used. The correspondingbuilt as described above is shown in Fig. 38.

Unfortunately, this model fail to produce good convergence in many caThis is not very surprising, as the nonideal wave two-ports introduce ational poles and zeros, while this model preserves the order of the filter.

Even in the cases where the optimization converge to good solutions, it tto generate different sets of component values which yields the same trafunction, even for small changes in magnitude of the active error parame

Fig. 39 shows how the component values of the model filter changes, wthe time-constant part of S22 of the wave two-port that implements C2 of theoriginal filter is multiplied by 1+[-16,-8,-4,-2, 2, 4,8,16]/1000 (G1 and G3 arethe unmarked curves which are very close to zero). For example, one cathat L1 increases when this time-constant is decreased, but remains constthe time-constant is increased. It is also interesting to note that C2 change lessthan most of the other component values, as this is the component immented by the wave two-port where the error is introduced.

This is one of the cases when the model works best, with good matchinthe transfer functions, with most values following some trends and few cawhere parameters have converged towards different solutions.

Figure 38: Simple LC-filter based network for modelling WAF errors

ε 1ε0-----max H0 ω( ) Hm ω( )–=

R'

R'

L1 L3 L5

C2

VIN U VOUT

L2

R3

R2

C4

L4

R4G1 G3 L

S


ltertorsanddur-

s or

Another model was created by changing the components in the original fiinto the nets shown in Fig. 40a and Fig. 40b, the former used for inducand parallel resonant circuits, while the later was used for capacitorsseries resonant circuits. Source and termination resistances were variableing the optimization, but no additional components were added in serieparallel to them.

The corresponding filter is shown in Fig. 41.

Figure 39: Results from modelling of WAF errors using the simple model

Figure 40: Component replacements for the extended model

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

com

pone

nt v

alue

error

L4

L3

C4

L2

L5

RS

L1

RL

L1

C2

R2

R4 R

2

R L

CR L C

(a) (b)


s thepti-es

1.3.1still

eralin

nt

theeres notrent

This model did converge better than the previous, which is reasonable, aorder of the model-filter has increased. However, the problem with the omization converging to different solutions for different error-magnitudremained, in some cases the differences were larger still.

The models were also tested on the 5th order Chebyshev filter used inand 1.3.2, with better worst-case results than with the Cauer filter, butfailing to produce useable data.

In an attempt to find solutions for the cases that failed to converge, sevinitial values of the optimization parameters were tested. This resultedimproved matching of the transfer functions, but the problem with differesets of solutions for different error magnitudes even larger.

Some attempts at increasing the order of the model further by extendingmodel with reactive elements at the source or termination of the filter wperformed. This improved the convergence in some cases, but this wainvestigated further as the resulting filter was considered to be too diffefrom the original filter to be useful for analysis.

Figure 41: Extended LC-filter based network for modelling WAF errors

R

R

L1 L3 L5

C2VIN U VOUT

L2

R5R1 R3

R2

C1 C3 C5

C4

L4

R4

Chapter 7 –Tuning Strategies for Wave Active Filters 51

ndsiteThis

stopthetheingt inther

.3.2

toer,tion.l ber to

lteralso

-fre-

f allouldin art at

isoulding,

easten-iveing-port

7 Tuning Strategies for Wave Active Filters

Unlike other active filter structures, wave active filters have two in-ports atwo out ports, with the desired transfer function realized between oppoports, and the complementary transfer function between adjacent ports.gives the opportunity to implement rather unique on-line tuning systems.

One possibility is to add a reference-signal to the input signal inside theband of the filter, if possible at a zero in the transfer function, and usereflected signal for tuning. Some means of distinguishing this signal fromsignal being filtered out are required, which makes the “correlated tunloop” described in 2.3.1 well suited, as one can now add signal to the inpuorder to ensure enough spectral contents for the loop to be stable. Anoalternative may be to use spread-spectrum techniques as described in 2

Another possibility is to add a reference-signal at the input port oppositethe normal input port, with a frequency well within the passband of the filtpreferably at a frequency where the transfer function has zero attenuaSince the filter has very little attenuation inside the passband, there wilvery little reflected signals at these frequencies, possibly making it easiedistinguish the reference-signal from signals filtered out in the main fipath. Using a reference-frequency inside the passband of the filter mayimprove the accuracy of the tuning, especially if more than one referencequency 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 othe time-constants present in the filter, another possibility arises; one cuse a single reference two-port and a single gradient generation circuitmodel-matching tuning scheme, as described in 3.2.5, tuning one two-pothe time. As with normal model-matching tuning, assumes that thereenough signal present in the filter when the tuning is performed, one shprobably detect the current level of input signal before attempting any tunto avoid detuning the filter during periods of silence.

Most standard on-line tuning schemes would of course also work, at lwith the original buffer-based implementation, where the high gain-error ssitivity of wave-active filters isn’t a problem, as they are relatively insensitto other errors. When other implementations are used, an off-line tunscheme would probably be necessary to ensure that the gain of each twois correct.


aveudethegnaloorak-ay

on-o al therateallyri-

One straight-forward method would be just feeding a signal through the wtwo-port, for example at it’s -6dB frequency, where the same signal amplitshould be output from both ports, when the filter is correctly tuned. Thengain can be adjusted until both levels are very close to the half the input siamplitude. However, designing a good peak-detector is non-trivial, and pperformance in tuning-circuits is often the result of badly designed pedetectors. With careful design and offset cancelling this simple method mstill be useable.

Another tuning strategy especially suitable for wave active filters is to recnect parts of the circuit into resonant circuits, which are then tuned tdesired center frequency. The gain of the two-ports can be adjusted untioscillation is stable in the linear range of the circuit, both ensuring accugain tuning and reducing the problem of frequency-tuning-errors that usuresult when the amplitude of the oscillation is limited only by the nonlineaties of the circuit.

Chapter 8 –Conclusions and Future Work 53

averm-r is

-cir-eingthen the

no, twocialble toput

rom

t thedicd in

d forandgcaseslter.

esstheits ison-

8 Conclusions and Future Work

8.1 Tuning of Continuous-Time Integrated Filters

While many techniques for tuning continuous-time integrated filters hbeen proposed, they all depend on the type of filter being tuned, the perfoance specification, and in many cases the environment in which the filtebeing used.

The more generic techniques for on-line tuning relay on using a referencecuit for measuring parameters relevant to the performance of the filter btuned, rather than actually measuring on the actual filter itself. Due tomismatch between circuits on the same chip, the achievable accuracy is iorder of 1%, strongly dependent on the filter being tuned.

On the other hand, if one wish to measure on the filter directly andassumptions can be made on the spectral contents of the input signalpossibilities remain. If the SNR required by the application is low, and spereference-signals or implementations are used, it is in some cases possihave the reference-signal passing through the filter together with the insignal, without excessive reduction of the SNR or severe interference fthe input signal on the tuning-circuit.

One alternative is to use an off-line tuning scheme, which assumes thafilter can be taken out of the signal processing path occasionally for periomeasurements and retuning (possibly with an identical filter used as stanwhile tuning is performed).

If the input signal can be assumed to have enough energy in the band usetuning, it is possible to measure the effects of the filter using this signal,use this information for tuning the filter. The tuning-circuit for performinthese measurements tend to become relatively complex, and in somehave a large area overhead due to the need for an additional reference fi

8.2 Wave Active Filters

While Wave Active Filters in their most basic implementation do possgood tolerance towards component variations, the implementation oftime-constant control necessary to make them useable in integrated circunot a trivial task. The attempts made in this thesis to find a more easily c


veble ift an

orks therea,

ple-ele-ing

trolled implementation, without significantly increasing the sensitivity hanot been successful. The studied structures may however still be useaadequate tuning-circuitry is used, but it is questionable if they represenimprovement over other structures such as leapfrog-filters.

Even though WAFs were first proposed almost 30 years ago, very little whas been performed implementing them. Furthermore, in these casefocus has mainly been on finding structures reducing the required circuit awith severely degraded performance as a result. A study of on-chip immentations using the original implementation using only buffers as activements may be interesting, especially in combination with one of the tunmethods suggested in chapter 7.

Chapter 9 –References 55

ta--

lsets,

ir-

s-ov

8-

id-

s-

s,

o-r-

9 References[1] Hastings, A.; “The Art of ANALOG LAYOUT,” Prentice Hall, 2001, ISBN 0-

13-087061-7.

[2] David, A. Johns; Ken, Martin; “Analog Integrated Circuit Design,” JohnWiley & Sons inc, 1996, pp. 626, ISBN 0-471-14448-7.

[3] Gopinathan, V.; Tarsia, M.; Choi, D.; “Design considerations and implemention of a programmable high-frequency continuous-time filter and variablegain amplifier in submicrometer CMOS,” IEEE Journal of Solid-State Cir-cuits, Volume: 34 Issue: 12 , Dec 1999, pp. 1698-1707.

[4] Dehaene, W.; Steyaert, M.S.J.; Sansen, W.;A “50-MHz standard CMOS puequalizer for hard disk read channels,” IEEE Journal of Solid-State CircuiVolume: 32 Issue: 7 , Jul 1997, pp. 977 -988.

[5] Stefanelli, B.; Kaiser, A; “A 2-µm CMOS fifth-order low-pass continuous-time filter for video-frequency applications,” IEEE Journal of Solid-State Ccuits, Volume: 28 Issue: 7 , Jul 1993, pp. 713-718.

[6] Yamamoto, T.; et. al; “FM audio IC for VHS VCR using new signal procesing,” IEEE Transactions on Consumer Electronics, Volume: 35 Issue: 4 , N1989, pp. 723-732.

[7] Silva-Martinez, Jose; Steyaert, Michael S. J.; Sansen, Willy; “A 10.7-MHz 6dB SNR CMOS continuous-time filter with on-chip automatic tuning,” IEEEJournal of Solid-State Circuits, v 27, n 12, Dec, 1992, pp. 1843-1853.

[8] Chang, Zhong Yuag; Haspeslagh, D.; Verfaillie, J.; “Highly linear CMOSGm-C bandpass filter with on-chip frequency tuning,” IEEE Journal of SolState Circuits, v 32, n 3, Mar, 1997, pp. 388-397.

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