Inter-Ing INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, November 2007.

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1 Inter-Ing 7 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 5-6 November 7. Bessel Filter implementation in Log-Domain ROBERT GROZA, LELIA FESTILA, ERWIN SZOPOS Technical University of Cluj-Napoca, Faculty of Electronics, Telecommunication and Information Technology str. Gh. Bariţiu, nr 6-8, 47, Cluj-Napoca Keywords: Log-Domain Integrators, Bessel Filter, low pass filter, linear group delay Abstract: In signal processing, mainly in audio domain, the need for a linear phase characteristic is sometimes more important than the frequency specifications concerning the pass band and stop bandwidth. For these applications, we need a Bessel approximation for the frequency characteristics. If the application is designed for portable devices low power and low voltage building blocks are also required. Log-Domain circuits may be very suitable. We propose and analyze in this paper a log-domain 4 th order Bessel filter and emphasize its performance in comparison with an AO-RC design.. Introduction The purpose of the Bessel filter is to achieve approximately a linear phase, this one being equivalent to a constant time delay. This is the best phase response from an audible standpoint, assuming you don t want to correct an existing phase shift. Bessel low-pass filters have a maximally flat group delay at about Hz, so the phase response is approximately linear in the pass-band, while at higher frequencies the linearity degrades, and the group delay drops to zero (see Fig. a) and b)) a) s... b) Fig. 4 th order Bessel filter characteristics: a) magnitude, b) group delay IV-3-

2 This non-linearity has a minimal impact because it occurs primarily outside the bandwidth when the output level is low. In fact, the phase response is so close to a time delay that Bessel low-pass and all-pass filters may be used only to produce a time delay. We designed a log-domain Bessel filter and analyze it in this paper in comparison with the LC prototype and an equivalent AO-RC variant.. Log-Domain integrator Log-domain integrators are the main building blocks in log-domain filtering technique. They are essentially composed of two push-pull exponential cells shown in Fig. By applying Kirchhoff Voltage Law across the translinear loops of these cells [], the output current is found to be: out o (Vin -V out ) VT I =I e () V out Q Q ut Q Q V in Q3 Q4 V in Q3 Q4 ut V out (a) Fig. Exponential cells with opposite polarities. a) positive exponential circuit, b) negative exponential circuit This cell is also known as voltage-programmable current mirror, in which the bias current is modulated by the difference of two voltages, V in and V out. Using a capacitor and the two log-domain circuits of Fig., a log-domain integrator can be formed, as shown in Fig. 3. (b) V out V ip C V in node A Fig.3 Log-domain integrator IV-3-

3 By applying Kirchhoff Current Law in node A, its transfer function results in the equation for ideal nonlinear log-domain integrator, depicted below: Exp(V )= I {Exp(V )-Exp(V )}dt () o o ip in VT C 3. 4 th order Bessel filter In Fig. 4 the LC low pass filter prototype is shown, that was designed using the normalized LC values for R g =Rs= from []. The values were frequency scaled to obtain a khz cut-off frequency. Rg L 3.7u L 7u _LPF_Pasiv Vin C u C 36u Rs Vout Fig.4 4 th order Bessel filter Starting from this circuit, we designed an AO-RC active Bessel filter. The AO-RC filter derived from the transfer function of the filter presented in Fig. 4 is a cascade of two second order multiple feedback filters (MFB). The second order cell schematic for a MFB low pass filter is shown in Fig. 5: R C Vin R R3 - Vout C + Fig. 5 Second order MFB low-pass filter We start the design with the transfer function that can be written []: A H( s) = (3) + as + bs R RR 3 where A =, a = ωcc R + R3+, b = ωccc RR3. R R Given ω c, Q, A and choosing arbitrary C C, we can obtain the values for the resistors R R3: IV-3-3

4 R ( ) ac a C 4bCC A = (4) 4π fcc c R R R = (5) A = b (6) 3 4π fc CC R The resulting active Bessel filter is presented in Fig. 6. Vin R3 88.7k R8 88.7k R k C4 6p C3 p V- U9 - AD74 OUT + V+ R5 9.47k R7 9.47k R6 45.3k C5 35p C6 68p V- U - AD74 OUT + V+ MFB Vout Fig. 6 4 th order active MFB Bessel Filter We compare this filter with the log-domain Bessel filter which was designed starting from the LC prototype and using the F - NF method presented in []. The Log-Domain filter presented in Fig. 8 is composed of four log-domain integrators, an input logarithmical cell and an output exponential cell. The integrator is the one presented in the previous section (Fig. 3). The logarithmical cell is given in Fig. 7 and is derived from a voltage programmable current mirror []. +V A +V A I x 3 4 V o The output voltage is of the form: -V A Fig. 7 Logarithmic circuit ( ) V = V ln I / I (7) o T x o IV-3-4

5 V+ DC = AC = TRAN = I U6 Log_I IO = u Iin Vout U N_I IOP = u ION = u C = 7.5p 3 u U I U N_I IOP = u ION = u C =.5p 3 U U3 N_I IOP = u ION = u C = 38.5p 3 U3 U4 N_I IOP = u ION = u C = 695.5p 3 U4 U5 ExP_I V+ 3 _LPF_LD Iout V- IO = u Fig. 8 4 th order log-domain Bessel filter 4. Simulation results The simulations were carried out with ORCD s PSpice simulator. A design example is presented for a log-domain Bessel low-pass filter with the performances: khz cut-off frequency at -3dB and 5kHz cut-off frequency at -5dB. Its characteristics were compared with the passive LC prototype and the derived from it classical active filter designed for the same performance. The LC filter was considered as the reference circuit. The amplitude characteristics, the group delay characteristics and unit step response of the filters have been considered in the analysis procedure. The simulation results are presented in Fig. 9 and Fig.. In Fig. 9 a) the magnitude characteristics are shown, the upper one belongs to the passive implementation and the lower to log-domain simulation. One can see that the log-domain filter cut off frequency is a little bit lower than the AO-RC-MFB implementation. This is due to BJT non-idealities, finite β and Early voltage [4][5]. In Fig. 9 b) one can see the group delay of the three filters. Fig. 8 shows the unit step responses. One can see the log-domain circuit responds faster than the other two prototypes, which is an important advantage of the log-domain implementation. The log-domain implementation is a current-mode implementation that makes possible a faster response in time domain, which is a benefit for many applications (filtering and control systems).. 4us -5 (KHz) - us -5 - db(v(out_lpf_pasiv)) s 4us G(V(OUT_LPF_PASIV)) -5 - (9.97KHz) us -5 - db(v(mfb)) s 4us G(V(MFB)) -5 - (9.6KHz) us -5 -.Hz Hz KHz.MHz db(i(_lpf_ldreal)) Frequency s Hz.KHz KHz KHz.MHz G(I(_LPF_LDReal)) Frequency a) b) Fig. 9 Simulation result for the presented filters: a) frequency characteristics, b)group delay IV-3-5

6 5mV 5mV V V(OUT_LPF_PASIV).V.5V V ua V(MFB) ua A s 5us us 5us us 5us I(_LPF_LDReal) Time Fig. Simulation result for the presented filters: unity step response 5. Conclusions Current-mode circuits have many advantages over the traditional AO-RC design regarding chip are, low voltage levels and speed. Using log-domain design for realizing Bessel filters may be a good alternative to classical variants. From the simulation results, one can see that the frequencies characteristics of the logdomain filter vary insignificant in comparison with the AO-RC filter s ones. Deviations in both characteristics are largely dependent on the BJT non-idealities. Further investigations are needed to reduce them. Analyzing the time response, one can see the main advantage of the log-domain design that is a faster response with a small value of the over-shoot parameter. References: [] Steve Winder Analog and digital filter design, Newnes, [] D. Frey Log-domain filtering: an approach to current mode filtering, IEEE Proceedings-G, vol. 4, no. 6, pp , Dec. 993 [3] L. Feştilă, M. Ţopa, S. Hintea, M. Cîrlugea, R. Groza, A general modular design of ELIN filters based on F - NF models A&QT-R IEEE-TTTC International Conference on Automation, Quality and Testing, May 4 [4] V.W. Leung, M. El-Gamal, and G.W. Roberts, Effects of transistor nonidealities on log-domain filters in Proc. IEEE Int. Symp. Circuits and Systems, June 997, pp. 9-. [5] V.W. Leung, and G.W. Roberts, Effects of Transistor Nonidealities on High-Order Log-Domain Ladder Filter Frequency Responses IEEE Trans. on Circuits and Systems-II: Analog and Digital Signal Processing, pp , May IV-3-6

Inter-Ing INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, November 2007.

Inter-Ing INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, November 2007. Inter-Ing 2007 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007. A FULLY BALANCED, CCII-BASED TRANSCONDUCTANCE AMPLIFIER AND ITS APPLICATION