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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 Avdelning, Institution Division, Department Datum Date Electronics Systems, Dept. of Electrical Engineering 581 83 Linköping 2003-06-05 Rapporttyp Report category ISBN Language Svenska/Swedish Licentiatavhandling ISRN × Engelska/English × Examensarbete Språk C-uppsats D-uppsats — LITH-ISY-EX-3401-2003 Serietitel och serienummer Title of series, numbering Övrig rapport ISSN — 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. 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