An Approach for Tuning High-Q Continuous-Time andpass Filters

K. A. Kozma, D. A. Johns and A. S. Sedra

University of Toronto Department of Electrical and Computer Engineering Toronto, Ontario, M5S 1A4, Canada karen @ eecg .toronto.edu (416) 978-3381

AB!STRACT A new technique for tuning high-Q continuous-time filters is presented. This technique is a modified version of an adaptive tuning method where the tuning signals are sinusoids generated by digital bandpass delta sigma oscillators. As will be shown, the required building blocks for the tuning system are simple. Simulation resiults are presented to show the feasibility of the proposed approach. INTRODUCTION
In 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 not necessarily exact after fabrication, hence, the need for tuning [ I]-[4]. This paper discuss(-s the tuning of continuous-time bandpass filters with a modified version of the adaptive tuning method [SI. Instead of using discrete-time tunling signals. sinusoidal signals are employed. It will be shown that these sinusoidal signals are easily generated using a digital bandpass delta sigma (AX) oscillator. This oscillator is simply a digital oscillator, the circuitry of which is greatly simplified by using a bandpass AX modulator. The AX modulator converts a multi-bit signal to a single-bit stream so that complex and time-consuming operations, such as multiplication, are simplified. The bandpass AX modulator lends itself nicely to this application as the inherent noise shaping, which occurs with this modulator, works well with high-Q bandpass filters. A:; will be shown, the tuning input and reference signals. gencmted by the proposed oscillator, are practically noiseless for frequencies within the passband of the filter; but the noise increases for frequencies further away from the passband. However, this out-of-band noise is not detrimental since for high-Q filters, this noise is significantly attenuated. Therefore, higher-Q bandpass filters are more attractive when using the bandpass AX oscillators. This paper begins by briefly reviewing the adaptive tuning system and describes the use of sinusoidal tuning signals. The concept is shown to be valid through a simulation example where a fourth-order bandpass filter. with a Q-factor of 27, is tuned. Next, thc proposed b:andpass AX oscillator is explained. Then, the generation of the sinusoidal tuning signals, using the bandpass AX oscillator, is described. Lastly, a simulation example is presentcd where the poles of the same fourth-order bandpass filter are tuned using the bandpass AX oscillator to generate the tuning signals.

Fig. 1 The adaptive tuning system.

THE, ADAPTIVE TUNING TECHNIQUE USING SINUSOIDAL TUNING SIGNALS
The adaptive tuning technique, which has been shown to be effective for tunin continuous-time low-pass filters is illustrated in Fi 1 , $asical!y, the adaptive approach requires two signals to Ee generated: a tuning input and a reference signal. The reference signal represents the output which the ideal filtcr would produce for, the 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 filter that determine the poles and zeros until the mean-squarederror value, MSE(e(t)), is minimized. The LMS algorithm uses the error signal, e(t), and a gradient sional, to tune the coefficients. The error si nal, is simply the dirference between the reference signal a n t the output of the filter. Since the structure chosen for the tunable filter, is an orthonormal ladder filter, it can be shown that the necessary gradient signals are easily generated by duplicating the tunable filter and swapping some coefficients [ 5 ] .Therefore, in order to make the system practical, simple tuning input and rcference signals should be devised. Although a pseudo-random se uence was used as the tuning input in previous instances [. 1 161, it is.not appropriate for tunin bandpass filters as it iri, spectrally rich over a laroe range o f frequencies. As a result, most of the u s e h information is severely attenuated by a bandpass filter. Thus an in ut signal that is spectrally rich in the band of interest, nameyy the passband, appears to be a better choice. For thc bandpass case, the authors suggest the use of a sum of sinusoids, the frequencies of which are placed within the passband of the desired transfer-functlon, so that its information is useful and not greatly attenuated. Given that the in U I is a sum of sinusoidal si nals, the reference signal wou1b)also be a sum of sinusoida? signals as the filter is a linear system. Consider, the maximally-flat fourth-order bandpass filter

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centered at 0.7588 radls and having a Q-factor of 27:

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an oscillator is required to generate the tuning signals.

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(s2+0O2O1s+O.S91O) ( . ~ + 0 . 0 1 9 6 s + O . S 6 0 9 )

(1)

BANDPASS DELTA SIGMA OSCILLATOR

Since both of the required signals for the adaptive tuning system are sinusoids, a bandpass AC oscillator may be used to generate both signals. A bandpass AX oscillator, which is based on a low-pass AZ oscillator [ 7 ] , depicted in Fig. 3. It is uses the same principle as the low-pass AX oscillator, the only difference being that the fourth-order bandpass AX modulator shown at (a) in Fig. 3, replaces a second-order low-pass AZ modulator. Hence, the bit stream produced by the modulator has a band-reject noise transfer-function instead of a high-pass noise transfer-function.
AX modulator

to be the ideal transfer-function. A plot of this transferfunction is shown in Fig. 2. If the following sum of five

bandpass

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(a) r,
single-bit output

-ideal --initial

Freq. [radk]

register 2 multi-bit single-bit

Fig. 2 Plot of ideal, initial and tuned transferfunctions using ideal oscillators.
sinusoidal waves: input = sin (0.7500t) + sin (0.7566r) sin (0.7588t) + sin (0.7634r) + sin (0.7667t) (2)

is the input to the filter in ( I ) . the ideal output can be easily calculated to also be a sum of five sinusoids at the same frequencies but with the following gain-factors and phase shifts:
jrdrui =

0.9283 sin (0.7S00t + 0.9805) t 0.9997 sin (0.7566t + 0.2229) + 1.0000sin ( 0 . 7 5 8 8 ~ 0.0000) + + 0.9945 sin ( 0 . 7 6 3 4 - 0.4748) ~ 0.9542sin (0.7667t - 0.8567)

Fig. 3 Digital bandpass &X oscillator. It can be shown that the frequency of oscillation is set by the product of the coefficients a1 and a2 and that the amplitude of oscillation is set by the initial value in register 1 [7]. Specifying the coefficient a2 to be a power of 2 makes this multiplication a simple shift operation and the difficult multiplication by the multi-bit coefficient, a l , is simplified as the bandpass AX modulator produces a single-bit output. Consequently, the multi-bit multiplication is reduced to a simple multiplexor as shown in Fig. 3. To simplify the required circuitry of the AZ modulator, the oscillator is defined to have its oscillation frequency around fs/4. In other words, the sampling rate of the oscillator is defined to be four times the passband of the filter. Also, a fourth-order bandpass AX modulator was chosen so that there is at least second-order noise-shaping. The AZ modulator used in this paper is illustrated in Fig. 4. Notice that the AX

+
,

(3)

Using the sum of sinusoids in (2) and (3) as the tuning input and reference signals respectively, and arbitrarily initializing the tunable filter to the transfer-function shown in Fig. 2 the adaptive tuning system was simulated. The coefficients which dctermine the poles and the finite zeros in the lower stopband were adapted while the two zeros at infinity were fixed. As expected the passband is accurately matched, however. the lower stopband i s not exact since the tuning input focused on frequencies within the passband. This simulation shows that the adaptive tuning system can accurately tune the passband of a bandpass filter when both the tuning input and reference signals are sinusoids. Therefore

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multi-bit single-bit

Fig. 4 Fourth-order bandpass AX modulator.

modulator consists of simple circuitry: delays, multi-bit adders and a latched comparator. A plot of the spectrum of the output of the bandpass AX oscillator, y s , is shown in Fig. 5. The frequency of oscillation here is fs14; this can be changed somewhat by changing but a l . However, the frequency of oscillation cannot be varied too

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Fig. 6 Simulation setup to determine the reference signall.

yret ( t ) = ay, + b,..
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Fig. 5 Spectrum of the output of the bandpass AX oscillator.

much as it becomes buried i n the noise added by the AZ modulator. Hence, the bandpass A 2 oscillator can be used to produce sinusoidal signals that are very close together in frequency which is exactly what is required for tuning high-Q bandpass filters.

(6)

GENERATION OF THE SINUSOIDAL TUNING SIGNALS For the fourth-order bandpass filter discussed in this paper, five bandpass AX oscillators are employed to generate five sinusoidal signals at different frequencies within the passband of the filter. Thus the tuning input is the a sum of the y, outputs of the bandpass AZ oscillators. Notice that this sum is simple as it is just the addition of five single-bit streams. Given this tuning input, a suitable reference signal must be generated. Although five osc~illatorsare used, consider the input of a single sinusoid. J', . As shown before. the ideal output should also bc a sinusoidal signal with a specific gain and phase shift. However, this is no1 exactly the case here as the AX modulators do add noise to the sinusoidal signals, so that j' can bc written as sin ( u t )+ e l ( t ) , where e , ( t ) is the n&e added to the pure sinusoidal wave. Assuming that e ( t ) is zero in the band of' interest, the output of the ideal filter, ! ' , d e a i ( t ) , can be shown to be ksin ( o t + A ) where k is thc gain and A is the phase shift of the filter at the frequency o rads. From trigonometry, this is equivalent to:
~

Now, only a and p have to be determined. Since is not exactly 90 degrees out of phase with y,, and p, and y , do have an added noise content, the calculation of a and p is not obvious. Hence the approach used in [ 5 ] , to determine the reference signal, is utilized here. As in [5], simulations are performed using the adaptive tuning system to tune the a and PI coefficients. The setup used in the simulations is illustrated in Fig. 6. In this case the reference signal is y r d e u i ,and the signal being adapted is y r e , . The coefficients being adjusted are a and @. As mentioned before, the LMS algorithm adjusts I:he coefficients by using the error signal, e(t), and gradient signals. The gradient signals are the gradients ayre,/aaand d j r L , / d p . It is obvious from ( 6 ) ,that these gradients are:

~ , ~ , , , ~ , (= ) ksin ( u t ) ( A ) + kcos (ut) sin ( A ) , t cos

(4)

where cos ( A ) and sin ( A ) are simply scalars. Denoting kcos ( A ) by a and ksin ( A ) by @, (4) can be rewritten as:

Hence, all the signals required for adapting the reference signals are readily available. The dotted boxes in Fig. 6, represent filters that are used in the :simulations to reduce the noise added by the AZ modulators and thus isolate the useful signals in the band of interest. This does not interfere with the tuning of cx and p as any gain or phase shift caused by the . filters is applied to both paths: yrc, and j i d e l ,Therefore, the filters only aid in speeding up thc simufations and their removal should not yield different results. For simplicity the transfer-function in (1) is used for T. Pcrforming this simulation on cach of the five bandpass AZ oscillators, used to generate the tuning input. give thc specific a 's and p's, required by (6). Therefore, only five bandpass AX oscillators are needed to create both the tuning input and reference signals. Fig. 7 illustrates sample waveforms of the tuning inpul. and reference signals.

It can be shown that the output. yc, at (b) i n Fig. 3 is approximatcly 90 degrees out of phase to J , Thus the c o j ( u t ) componcnt of (3, be obtained from y , so that can the reference signal can be generated as follows

EXAMPLE:
A simulation example was performed to validate thc use of

sinusoidal signals, generated by banidpass AZ oscillators. for tuning the passband of the fourth-or'der bandpass filter. Since the adaptive algorithm places more emphasis on tuning the

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(a) Tuning input.

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I, ; ;
.-. tuned

n
U

Freq. [rad/s]

Fig. 8 Plot of ideal, initial and tuned transferfunctions using bandpass AX oscillators.
filter. Furthermore, it has been shown that the circuitry required to realize these bandpass AX oscillators is simple. Simulation results were presented to show the feasibility of the proposed tuning approach.

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CONCLUSIONS

REFERENCES
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(b) Reference signal.

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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 the filter are tuned in this example. Also, it is not expected to have accurate stopband matching as the reference signal is only valid for frequencies within the passband of the filter. Any information at frequencies in the stopband of the filter is buried in noise added by the AZ modulators. The ideal, initial and tuned transfer-functions are shown in Fig. 8. As in the previous example, the same ideal transferfunction is used and the tunable filter is similarly initialized. After simulating the adaptive tuning system where both the tuning input and reference signals are generated by five AZ oscillators the filter was tuned to the transfer-function shown in Fig. 8. Notice the accurate matching achieved in the passband. A modified version of the adaptive tuning system was presented for tuning high-Q continuous-time bandpass filters. Instead of using discrete-time tuning input and reference signals, sinusoidal signals that are generated by bandpass AX oscillators are employed. The inherent noise shaping, due to the bandpass AX oscillators, works well with high-Q bandpass filters as they are used to produce sinusoidal signals close together in lrequency and well within the passband of the

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J. Silva-Martinez, M. S. J. Steyaert and W. Sansen, A 10.7-MHz. 68-dB SNR CMOS Continuous-Time Filter with On-Chip Automatic Tuning, IEEE Journal of Solid-state Circuits, vol. 27, pp. 1843-1853, Dec. 1992. G. Groenewold, A High-Dynamic-Range Integrated Continuous-Time Bandpass Filter, IEEE Journal of Solid-state Circuits, vol. 27, pp. 1614-1622, Nov. 1992. Y. Wang and A. A. Abidi, CMOS Active Filter Design at Very High Frequencies, IEEE Journal of SolidStale Circuits, vol. 25, pp. 1562-1573, Dec. 1990. C. Chiou and R. Schaumann, Design and Performance of a Fully Integrated Bipolar 10.7-MHz Analog Bandpass Filter, IEEE Transactions on Circuits and System. vol. 33, pp. 116-124, Feb. 1986. K. A. Kozma, D. A. Johns and A. S. Sedra, Automatic Tuning of Continuous-Time Integrated filters Using an Adaptive Filter Technique, IEEE Transactions on Circuits and Systems, vol. 38, pp. 1241-1248, NOV. 1991. K. A. Kozma, D. A. Johns and A. S. Sedra, Tuning of Continuous-Time Filters in the Presence of Parasitic Poles, IEEE Transactions on Circuits and Systems, vol. 40, pp. 13-20, Jan. 1993. A. K. Lu, G. W. Roberts and D. A. Johns, A HighQuality Analog Oscillator Using Oversampling D/A Conversion Techniques, IEEE Transactions on Circuits and Sj,stems, vol. 41, pp. 437-444, July 1994.

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