8/2/2019 00519944

1/4

An Approach forTuning High-Q Continuous-Time andpass Filters

K. A. Kozma, D. A. Johns and A. S . SedraUniversity of TorontoDepartment of Electrical and Com puter EngineeringToronto, Ontario, M5S 1A 4 , C a n a d akaren @eecg toronto.edu (416) 978-3381

AB!STRACTA new technique for tuning high-Q continuous-time filtersis presented. This technique is a modified version of anadaptive tuning method where the tuning s ignals are s inusoidsgenerated by digital bandpass delta sigma oscillators. A s willbe shown, the required building blocks fo r the tuning systemare s imple. Simulation resiults are presented to show thefeasibility of the proposed approach.

INTRODUCTIONIn many communication channels (particularly wireless)there is a need for accurate high-Q bandpass filters . However,as with any integrated filter, the transfer-function is notnecessarily exact after fabrication, hence, the need for tuning[ I]-[4]. This paper discuss(-s the tuning of continuous-timebandpass filters with a modified version of the adaptive tuningmethod [SI. Instead of usin g discrete- time tunling signals.s inusoidal s ignals are employed. It will be shown that thesesinusoidal signals are easily generated using a digitalbandpass delta si gm a (AX) oscillator. Thi s oscillator is simplya digital oscillator, the circuitry of which is greatly simplifiedby using a bandpass AX modulator. The AX modulatorconverts a multi-bit signal to a single-bit stream so thatcomplex and time-consuming operations, such as

multiplication, are s implified. The bandpass AX modulatorlends itself nicely to this application as the inherent noiseshaping, which occurs with this modulator, works well withhigh-Q bandpass filters. A:; will be shown, the tuning inputand reference signals. gencmted by the proposed oscillator,are practically noiseless fo r frequencies within the passbandof the filter; but the noise increases for frequencies furtheraway from the passband. However, this out-of-band noise isnot detrimental s ince for high-Q filters , this noise iss ignificantly attenuated. Therefore, higher-Q bandpass filtersare more attractive when using the bandpass AX oscillators.This paper begins by briefly reviewing th e adaptive tuningsystem and describes the use of s inusoidal tuning s ignals . Theconcept is shown to be valid through a s imulation examplewhere a fourth-order bandpass filter. with a Q-factor of 27, istuned. Next, th c proposed b:andpass AX oscillator is explained.Then, the generation of the s inusoidal tuning s ignals , usingthe bandpass AX oscillator, is described. Lastly, a simulationexample is presentcd w here the poles of the sam e fourth-orderbandpass filter are tuned using the bandpass AX oscillator togenerate the tuning s ignals .

IFig. 1 The adaptive tuning system.

THE, ADAPTIVE TUNING TECHNIQUE USINGSINUSOIDAL TUNING SIGNALS

The adaptive tuning technique, which has been shown to beeffective for tunin continuous-time low-pass filters isillustrated in Fi 1 , $asical!y, the adaptive approach requirestwo signals to Ee generated: a tuning input and a referencesignal. The reference s ignal represents the output which theideal filtcr would produce for, he given tuning input. As in [ 5 ] .the adaptive a1 orithm used is the least-mean;squared (LMS)algorithm. The% MS algorithm adjusts coefficients of the filterthat determine the poles and zeros until the mean-squared-error value, MSE(e(t)), is minimized. The LM S algorithmuses the error s ignal, e(t), and a gradient s ional, to tune thecoefficients . The error si nal, is s imply the dirference betweenthe reference s ignal a n t the output of the filter. Since thestructure chosen fo r the tunable filter, is an orthonor mal ladderfilter, it can be shown that the necessary gradient signals areeasily generated by duplicating the tunable filter and swappingsome coefficients [ 5 ] .There fore, in order to make the systempractical, simple tuning input and rcference signals should bedevised.

Although a pseudo-random se uence was used as thefor tunin bandpass filters as it iri, spectrally rich over a laroerange o f frequencies . As a resu l t, most o f the u s e hinformation is severely attenuated by a bandpass filter. Thusan in ut signal that is spectrally rich in the band of interest,nameyy the passband, appears to be a better choice. For th cbandpass case, the authors suggest the use of a sum ofsinusoids, the frequencies of which are placed within thepassband of the desired transfer-functlon, so that itsinformation is useful and not greatly attenuated. Given thatthe in U I is a sum of s inusoidal s i nals , the reference s ignalwou1b)also be a sum of sinusoida? signals as the filter is alinear system.Consider, the maximally-flat fourth-order bandpass filter

tuning input in previous instances [.? 1 161, it is.not appropriate

0-7803-2570-2195 $4.00 0 1 9 9 5 IEEE 1037

http://toronto.edu/http://toronto.edu/

8/2/2019 00519944

2/4

centered at 0.7588 radls an d having a Q-factor of 27:7 (1)7.8609 ( .?( s2 + 0O 2 O 1s + O .S 9 1O ) ( . ~ + 0 . 0 1 9 6 s + O . S 6 0 9 )rid

8/2/2019 00519944

3/4

7-2 0z9 40

f44 f 2 3f44Freq.-1,

Fig. 5 Spectrum of the output of the bandpassAX oscillator.much as it becomes buried i n the noise added by the AZmodulator. Hence, the bandpass A 2 oscillator can be used toproduce sinusoidal signals that are very close together infrequency which is exactly what is required for tuning high-Qbandpass filters.

GENERATIONOF THE SINUSOIDAL TUNINGSIGNALS

For the fourth-order bandpass filter discussed in this paper,five bandpass AX oscillators are employed to generate fivesinusoidal signals at different frequencies within the passbandof the filter. Thus the tuning input is the a sum of the y,outputs of the bandpass AZ oscillators. Notice that this sum issimple as it is jus t the addition of five single-bit streams.Given this tuning input, a suitable reference signal must begenerated. Although five osc~illatorsare used, consider theinput of a single sinusoid. J' , . As shown before. the ideal

output should also bc a sinusoidal signal with a specific gainand phase shift. However, this is no1 exactly the case here asthe AX modulators do add noise to the sinusoidal signals, sothat j' can bc written as sin ( u t ) e l ( t ) where e , ( t ) isthe n&e added to the pure sinusoidal wave. Assuming thate ( t ) is zero in the band of' interest, the output of the idealfilter, ! ' , d e a i ( t ) can be shown to be ks in ( o t + A ) where kis thc gain an d A is the phase shift of the filter at the frequen cyo ads. From trigonometry, this is equivalent to:

~ , ~ , , , ~ , ( t )ksin ( u t ) os ( A ) + kcos (u t ) in ( A ) , (4)where cos ( A ) and sin ( A ) are simply scalars. Denotingkcos ( A ) by a and ksin ( A ) by @, (4 ) can be rewritten as:

I t can be shown that the output . yc, at (b) i n Fig. 3 isapproximatcly 90 degrees ou t of phase to Thus thec o j ( u t ) omponcnt of (3 , an be obtained from y , so thatthe reference signal can be generated as followsJ ,

v W T -,,

Fig. 6 Simulation setup to determine thereference signall.

y re t( t ) = ay, + b,.. (6)Now, only a an d p have to be determined. Since )) is no texactly 90 degrees ou t of phase with y,, an d p, an d y , dohave an added noise content , the calculat ion of a an d p is notobvious. Hence the approach used in [ 5 ] , o determine thereference signal , is utilized here.

A s in [5], simulations are performed using the adaptivetuning system to tune the a an d PI coefficients. The setupused in the simulations is illustrated in Fig. 6.In this case thereference signal is y r d e u i , nd the signal being adapted is y r e , .The coefficients being adjusted are a an d @ . As mentionedbefore, the LM S algorithm adjusts I:he coefficients by usingthe error signal, e(t), and gradie nt signals. Th e gradient signalsare the gradients ayre,/aa nd d j r L , / d p . t is obvious from( 6 ) , hat these gradients are:

Hence, all the signals required for adapting the referencesignals are readily available. The dotted boxes in Fig. 6,represent filters that are used in the :simulations to reduce thenoise added by the AZ modulators and thus isolate the usefulsignals in the band of interest. This does no t interfere with thetuning of cx an d p as any gain or phase shift caused by thefilters is applied to both paths: yrc, an d j i d e l , . herefore, thefilters only aid in speeding up thc simufations and theirremoval should no t yield different results. For simplicity thetransfer-function in (1) is used for T.Pcrforming this simulation on cach of the five bandpass AZoscillators, used to generate the tuning input. give thc specifica 's an d p ' s , required by (6).Therefore, only five bandpassAX oscillators are needed to create both the tuning input andreference signals. Fig. 7 illustrates sample waveforms of the

tuning inpul. and referen ce sign als.EXAMPLE:

A simulat ion exam ple was performed to validate thc use ofsinusoidal signals, generated by banidpass AZ oscillators. fortuning the passband of the fourth-or'der bandpass filter. Sincethe adaptive algori thm places more emphasis on tuning the

1039

8/2/2019 00519944

4/4

L

(a) Tuning input.

LJl

n

I

n

(b) Reference signal.

Fig. 7 Tuning input and reference signals.passband and hence the poles of a filter, for a white noise input[6]. only the coefficients that directly affect the poles of th efilter are tuned in this example. Also, it is not expected to haveaccurate stopband matching as the reference signal is onlyvalid for frequencies within the passband of the filter. Anyinformation at frequencies in the stopband of the filter isburied in noise added by the AZ modulators.

The ideal, initial and tuned transfer-functions are shown inFig. 8. As in the previous example, the same ideal t ransfer-function is used and the tunable filter is similarly initialized.After simulat ing the adaptive tuning system where both thetuning input and reference signals are generated by five AZoscillators the filter was tuned to the transfer-function shownin Fig. 8. Notice the accurate matching achieved in thepassband.CONCLUSIONS

A modified version of the adaptive tuning system waspresented for tuning high-Q continuous-t ime bandpass f i lters.Instead of using discrete-t ime tuning input and referencesignals, sinusoidal signals that are generated by bandpass AXoscil lators are employed. The inherent noise shaping, due tothe bandpass AX oscil lators, works well with high-Q bandpassfilters as they are used to produce sinusoidal signals closetogether in lrequency and well within the passband of the

magnific

I

9 F ~ I;;,I l(, .-. uned

Freq. [rad/s]Fig. 8 Plot of ideal, initial and tuned transfer-functions using bandpass AX oscilla-tors.

filter. Furthermore, it has been shown that the circuitryrequired to realize these bandpass AX oscillators is simple.Simulat ion results were presented to show the feasibi l i ty ofthe proposed tuning approach.

i l l

121

[31

i41

151

i61

[71

REFERENCESJ. Silva-Martinez, M. S . J. Steyaert and W. Sansen, A10.7-MHz. 68-dB S NR CM OS Cont inuous-Time Fi lt erwith On-Chip Automatic Tuning, IEEE Journal ofSolid-state Circuits, vol. 27, pp . 1843-1853, Dec. 1992.G . Groenewold, A High-Dynamic-Range IntegratedContinuous-Time Bandpass Fil ter , IEEE Journal ofSolid-state Circuits, vol. 27, pp. 1614-1622, Nov.1992.Y. Wang and A . A. Abidi , CM OS Active Fil ter Designat Very High Frequencies, IEEE Journal o Solid-Stale Circuits, vol. 25 , pp . 1562-1573, Dec. 1990.C. Chiou and R. Schaumann , Design and Per formanceof a Fully Integrated Bipolar 10.7-MHz AnalogBandpass Filter, IEEE Transactions on Circuits andS y s t e m . vol. 3 3 , p p . 116-124, Feb. 1986.K . A. Kozma, D. A. Johns and A . S . Sedra, AutomaticTuning of Continuous-Time Integrated f i lters Using anAdaptive Fil ter Technique, IEEE Transactions onCircuits an d Systems, vol. 38, pp . 1241-1248, NOV.1991.K. A. Kozma, D. A. Johns and A. S. Sedra , Tuning ofContinuous-Time Filters in the Presence of ParasiticPoles, IEEE Transactions on Circuits and Systems,vol. 40, pp . 13-20, Jan. 19 93.A. K. L u , G . W. Roberts and D. A. Johns, A High-Quali ty Analog Oscil lator Using Oversampling D/AConversion Techniques, IEEE Transactions onCircuits and Sj , s t ems , vol. 41, pp . 437-444, July 1994.

1040