Huertas Díaz, José Luis

Pérez Verdú, Belén

Rodríguez Vázquez, Ángel Benito

English

idUS. Depósito de Investigación Universidad de Sevilla

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 4, APRIL 1987 441 .- / I / I Fig. 6. Total frequency magnitude response of four 2-D QMF filters summa- tion. The values are nearly constant, i.e., (0.999821-1.000179035) or (- 0.000777 dB - 0.000777 dB). Fig. 7. The reconstruction error of Fig. 6 in enlarged scale. The largest error is located at the four cnmers of the 2-D QMF filters (0.999821-1.000179035) or (-0.000777 dB - 0.000777 dB). nal signal can be proved to cancel each other, as long as the F,( .Zi)‘s are symmetric half-band filters. We can then perfectly reconstruct the original signal without any distortion. The fac- torization of the F,( Zi)‘s into their maximum phase components H,,,, ( Z, ) and minimum phase components Hmin( Z,) is the same as in 1-D cases [lo]-[12]. IV. EXPERIMENTAL RESULTS Fig. 4 shows the four-band partition of the frequency spectrum in our 2-D subband coding scheme. First, we use the Parks- McClellan FIR filter program to design a 63-point symmetric half-band filter Fo(Zi), and the procedure in [12] is’adopted to factor out F, (Z,) into its 32-point maximum-phase H,, (Z,) and minimum-phase part H,,(Z,), respectively (see Tables I-III). The four 2-D QMF filters can be efficiently designed by the separable products of four 1-D QMF filters as in Fig. 3. The frequency magnitude responses of four 2-D 32 X 32 QMF filters are shown in Fig. 5(a)-(d). The reconstruction error is within + 0.000777 dB (see Figs. 6 and 7), and the largest error is located at the four comers of 2-D QMF filters, mainly due to the factorization by homomorphic cepstrum. V. CONCLUSIONS In this paper, a new 2-D subband coding scheme is proposed, which adopts the 1-D QMF filter technique and Vetterli’s sep- arable scheme. Two-dimensional QMF filters can be efficiently designed by a separable product of identical 1-D QMF filters; this new 2-D subband coder can exactly reconstruct the original, free of any distortion. This will be very useful for image coding and transmission applications. REFERENCES 111 121 [31 [41 [51 161 [71 181 [91 1101 1111 1121 N. S. Jayant and P. NOB, Digital Coding of B’aoeforms. Englewood Cliffs, NJ: Prentice-Hall, 1984, ch. 11, pp. 486-509. D. Esteban and C. Galand, “Application of quadrature mirror filters to split band voice coding schemes,” in Proc. 1977 IEEE Int. Conf. Acowt., Speech, Signal Process., (Hartford, CT), May 1977, pp. 191-195. M. J. T. Smith and T. P. Barnwell, HI, “A procedure for designing exact reconstruction filter banks for tree-structured subband coders,” in Proc. 1984 IEEE Int. Conf. Acourt., Speech, Signal Process. (San Diego, CA), Apr. 1984, pp. 27.1.1-27.1.4. M. J. T. Smith and T. P. Barnwell, III, “Exact reconstruction techniques for tree-structured subband coders,” IEEE Trans. Acourt., Speech, Sig- nal Process., vol. ASSP-34, no. 3, pp. 434-441, June 1986. F. Mintzer, “Filters for distortion-free two-band multirate filter banks,” IEEE Tram. Acoust., Speech, Signal Process., vol. ASSP-33, no. 3, pp. 626-630, June 1985. M. Vetterli, “Multi-dimensional sub-band coding: some theory and algorithms,” Signal Process., vol. 6, pp. 97-112, Apr. 1984. J. W. Woods and S. D. O’Neil, “Sub-band coding of images,” IEEE Trans. Acourt., Speech, Signal Process., vol. ASSP-34, no. 5, pp. 1278-1288, Oct. 1986. R. E. Crochiere and L. R. Rabiner, Multirate digital signal processing. Englewood Cliffs, NJ: Prentice-Hall, 1983, pp. 155-157. F. Mintzer, “On half-band, third-band, and N&band FIR filters and their design,” IEEE Trans. Acourt., Speech, Signal Process., vol. ASSP- 30, no. 5, pp. 734-738, Oct. 1982. R. Boite and H. Leich, “A new procedure for the design of high order minimum phase FIR digital or CCD filters,” Signal Process., vol. 3, pp. 101-108, Apr. 1981. G. A. Mian and A. P. Namer, “A fast procedure to design equiripple minimum-nh& FIR filters.” IEEE Trans. Circurts Svst.. vol. CAS-29. pp. 327-331, May 1982. , S. C. Pei and S. T. Lu, “Design of minimum phase FIR digital filter by differential cepstrnm,” IEEE Tram. Circuits Syst., vol. CAS-33, pp. 570-576, May 1986. High-Frequency Design of the Wien-Bridge Oscillator Using Composite Amplifiers A. RODRIGUEZ-VAZQUEZ, .I. L. HUERTAS, AND B. PEREZ-VERDU I. INTRODUCTION The finite gain-bandwidth (GB) product of the operational amplifier (OA) severely degrades the high-frequency performance of the conventional one-OA realization of the Wien-bridge oscil- lator [l]. In [l], Budak and Nay have shown that the GB influence can be reduced by the expedient of using two-OA composite amplifiers. In particular, these authors focused on a realization of the Wien bridge using Reddy’s amplifier [2] and developed an innovative design procedure that makes it possible to extend the range of operation of the oscillator. However, in spite of the advantages of using the amplifier in [l], there still Manuscript received December 15, 1986. The authors are with the Departamento de Electricidad y Electronica, Facultad de Fisica, Universidad de Sevilla, 41012 Sevilla, Spain. IEEE Log Number 8613333. 0098-4094/87/0400-0441$01.00 01987 IEEE Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 02,2020 at 15:23:52 UTC from IEEE Xplore.  Restrictions apply. 442 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 4,APRlL 1987 remains a strong dependence between the frequency of oscillation and the GB value (see [l, eq. (S)]). Furthermore, the analysis in [l] relies on linear models, thus disregarding any influence due to either the amplitude-controlling mechanism or the OA nonlinear- ities. In particular, the influence of the slew rate is very strong at high frequencies, and as a result, important disparities appear between theoretical and actual behaviors [3]. The circuit in this letter is intended to overcome the above drawbacks. We consider a composite-amplifier realization of the Wien-bridge oscillator and use a nonlinear analysis technique for modeling the influence of the slew rate on the circuit perfor- mance. We then show that, by properly designing the amplifier, it is possible to obtain high-frequency, lowidistortion sinusoidal signals whose ,frequency of oscillation is virtually independent of GB. Experimental data confirming these predictions are in- cluded. II. SLEW-RATEOSCILLATORMODEL The Wien-bridge oscillator circuit is shown in Fig. 1. Two different blocks can be distinguished, namely an amplifier and a passive RC network. The transfer function for this latter network is TK(S) = SWa S2 + S3Wa + Wa* where Wa = 1/RC. If an ideal amplifier were available, its gain would have to be fixed to a value of 3, thus making the circuit oscillate at a frequency Wo = Wa. However, in practice this idealized picture is only valid .at very low frequencies. When the frequency is in- creased, both the required gain value and the oscillation frequency deviate from those ideal values. Moreover, the influence of the amplifier nonlinearities gives rise to the onset of distortion. Clearly, all previous effects would have to be considered for a proper design, this requiring a careful choice of the amplifier. Let us consider that the amplifier in Fig. 1 is realized using the two-OA composite structure of Fig. 2 [4]. The high-frequency nonlinear behavior of this amplifier can be accurately analyzed by resorting to the use of the OA model shown in Fig. 3 [6]. Assume that each OA in Fig. 2 is replaced by such a model. In the frequency domain, we may assume the signal Vi to be sinusoidal and then analyze the circuit by using the describing- function technique. As a preliminary step, we have to identify which of the two OA nonlinearities is the dominant one. Obvi- ously, it will depend on the actual values of both the K and the P parameter in pig. 2. However, for the values to be considered herein, it can be shown that the nonlinearity of OA2 is the dominant one [3]. Taking this into account, we get the following transfer function for the amplifier in Fig. 2: s 1+&p G(S,A) =K s K s2 Kp (2) ‘+z N(A) +f N(A) where GB denotes the gain-bandwidth product of the OA (we have assumed GB, = GB,) and A is the-amplitude of the funda- mental harmonic at the input of OA2. The describing function is given’ by N(A) = (z/n)[sin-l(G/A)+(G/A)II~] (3)for (S/A) <l 11 elsewhere. Fig. 1. Block diagram of the Wien-bridge oscillator. R~/R, = P - 1 ; k/R3 * KP - I Fig. 2. Two-OA composite amplifier realization Fig. 3. Nonlinear model for the OA. Consider now that we are using the previous amplifier model in Fig. 1. From the theory of nonlinear systems [7], it is known that there is a limit cycle oscillation for every solution of &LWk(S)Is=,wo =l. (4 Using (2) in (4), we get the following characteristic equation: s4 Kp GB*N( A) + s3 K 3 WaKP GBN( A) + GB2N( A) 1 2 1+ [ 3WaK Wa’KP WaKP ss GBN(A) + GB*N(A) GB .I Wa*K +s 3wa + GBN( A) -KWa +Wa2=0 1 (5) which, after some algebraic manipulation, can be shown to exhibit the following solution: Wo=Wa 3 K= 1+9[ Wa/GBN(A)]* 2 ‘= N;A). This is a very interesting solution in that the actual frequency of oscillation is equal to the ideal one. As a matter of fact, given a value of the Wa of the &C network, previous equations give us the required values of K and P for achieving the goal of making Wo = Wa. However, in order to solve these equations, prior knowledge of N(A) is required. In practice, we may resort to the use of some distortion measure related to N(A). In particular, we propose to use the ratio (expressed in decibels) of the amplitudes Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 02,2020 at 15:23:52 UTC from IEEE Xplore.  Restrictions apply. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS,VOL.CAS-34,NO,4,APPJLl987 443 K TABLE I 2 .8 2.6 2.4 2.2 100 150 200 (a) Wo/Zn, Khz P 3.5 I I I -3OdB 3.3 - 3.1 - -4OdB -5OdB 31, I I I....I....I*...I 10 SO 100 150 200 (b) Wo/Za, Khz Fig. 4. (a) Plot of K versus oscillation frequencies for different distortion levels. (b) Plot of P versus oscillation frequencies for different distortion levels. of the third and the fundamental harmonic at the output of OA2 (this is meaningful because the nonlinearity we are dealing with is odd). Such a ratio can be calculated from the corresponding value of N(A), as in [7]. Using the results of these calculations in (6), we can obtain the plots in Fig. 4. There we show K versus Wo (Fig. 4(a)) and P versus Wo (Fig. 4(b)) for different values of the distortion parameter. Given Wo and the maximum distor- tion level, we can then obtain the required values of K and P. At this point, the only question which remains to be answered is whether or not the limit cycle oscillations are stable. Applying Loeb’s criterion [7], we obtain the following condition for stabil- ity [3]: Wa* -5- GB2 KP*+9-3K<O which can be shown to be fulfilled for the values given in (6). III. DISCUSSION OF RESULTS The design procedure introduced in the paper has been tested via experimental measurements. Table I is a summary of the‘ results we have obtained using LM747 dual operational ampli- EXPERIMENTAL RESULTS FOR DIFFERENT OSCILLATION FREQUENCIES Theoretical Experimental Wu/2n, kHz K wo/2n, kJ+z Dist., dB 22.7 2.98 22.5 -42 49.1 2.89 49.2 -41.5 88.4 2.68 88.3 -41 107.8 2.55 105.5 -37 154.8 2.2 147.7 -36 fiers and passive components with a l-percent tolerance. The first part of the table shows the theoretical values obtained during the design phase. These values have been calculated from (6) taking GB = 800 kHz and assuming the distortion to be -40 dB. The second part shows the measured parameters. As can be seen, there is very close agreement between each value of Wa and the corresponding frequency of oscillation. In fact, the deviation is less than 2 percent for frequencies up to 100 kHz. It compares favorably with the model proposed in [l]; there, the deviation is 2 percent for a, frequency less than 6 kHz (see [l, eq. (18)]). As a final point, it is important to mention that the proposed modeling technique is not directly applicable to a composite- amplifier structure different from the one in Fig. 2. This is because instability problems can appear. In particular, such problems arise when trying to use Reddy’s amplifier [3]. In fact, the composite amplifier in Fig. 2 has been selected from the whole cat&log of amplifier structures [4], [5] as the one best tailored for getting Wo = Wa [3]. [II [21 [31 [41 PI [61 [71 REFERENCES A. Budak and K. Nay, “Operational amplifier circuits for the W&-bridge oscillator,” IEEE Trans. Circuits Syst., vol. CAS-28, pp. 930-934, Sept. 1981. M. A. Reddy, “Operational-amplifier circuits with variable phase shift and their application to high-Q active RC-filters and RC-oscillators,” IEEE Trans. Circuits Syzt., vol. CAS-23, pp. 384-389, June 1976. B. Perez-Verdu, “Nonlinear modeling of operational amplifiers based RC-oscillators,” (in Spanish), Ph.D. dissertation, University of Sevilla, 1985. A. S&man, “Classification and generation of active compensated non- inverting VCVS building blocks,” Innl. J. Circuit Theory Appl., vol. 8, pp. 395-405, Oct. 1980. J. L. Huertas and A. Rodriguez-Vazquez, “On the active compensation of operational amplifiers based VCVS,” IEEE Trms. Circuits Syst., vol. CAS-29, pp. 497-506, July 1982. P. Allen, “A model of slew-induced distortion in single-amplifier active filters,” IEEE Trans. Circuits Syst., vol. CAS-25, pp. 565-572, Aug. 1978. A. Gelb and W. VanderVelde, Multiple Input Describing Functions and Nonlinear System Design. New York: McGraw-Hill, 1968. Delay-Time Sensitivity in Linear RC Tree NAVNEET K. JAIN, V. C. PRASAD, AND A. B. BHATTACHARYYA Abstrucf -The study of RC networks is important to understand digital MOS integrated circuits. Several authors studied these networks from the point of view of bounds on voltage waveform [l], [2], signal delay [3], etc. Manuscript received June 11, 1986. N. K. Jain and V. C. Prasad are with the Electrical Engineering Department, Indian Institute of Technology, Hanz Khas, New D&i-110016, India. A. B. Bhattacharyya is with the Centre for Applied Research in Electronics, Indian Institute of Technology, Hanz Khas, New Delhi-110016, India. IEEE Log Number 8611587. 0098-4094/87/0400-0443$01.00 01987 IEEE Authorized licensed use limited to: Universidad de Sevilla. Downloaded on April 02,2020 at 15:23:52 UTC from IEEE Xplore.  Restrictions apply. 

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High-Frequency Design of the Wien-Bridge Oscillator Using Composite Amplifiers

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High-Frequency Design of the Wien-Bridge Oscillator Using Composite Amplifiers

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