A new method for the implementation of arbitrary ... - Semantic Scholar

The Tau-Cell: A New Method for the Implementation of Arbitrary Differential Equations Andre van Schaik and Craig Jin

School of Electrical and Information Engineering University of Sydney Sydney, NSW 2006, Australia

ABSTRACT In this paper we present the tau-cell, a building block based on a log-domain low-pass filter, which allows the creation of arbitrary differential equations using a cascade of these cells in a manner similar to the state variable methods of filter realization. In the tau-cell implementation, however, all feedback is local, i.e., each first-order section only feeds back to the previous firstorder section in the cascade.

1. INTRODUCTION Most modem day digital systems, such as computers and telecommunication devices contain analogue continuoustime filters as necessary components [I]. These filters remove unwanted frequencies from a continuous-time real-world signal before converting the signal into the digital domain. Performing the filtering in the analogue domain before A/D conversion reduces the demands on the AID conversion and makes these systems easier to implement. Demanding analogue filters, however, have been notoriously difficult to implement on chips. This often makes necessary the use ofoff-chip filters and results in penalties in terms of subsystem board size, number of interfacing inputloutput pins, additional overhead circuits, etc. Ultimately, this increases system cost and power, and lowers design flexibility.. However, there has been intensive global research into full on-chip integration of analogue filters over the last 20 years, with significant results. Four main techniques are currently used: ( I ) the transcnnductance-capacitance(GM-C) filters, ( 2 ) active RC filters, (3) MOSFET-C filters, and (4) G A - O p A m p filters [I]. The most significant problem of all these filter types is their high power dissipation. A more recent technique, called log-domain filters, uses an intemally non-linear representation of the signal to operate at low supply voltages and thus reduce power consumption.

0-78U3-7761-31U31S17.UU82003 IEEE

The area of log-domain filtering has seen a number of filter synthesis techniques appear in recent years. The first was the state-space synthesis method developed by Frey from his original Log-domain paper for bipolar technology [2] and its adaptation for weak-inversion MOS technology by Toumazou [3]. Various synthesis techniques followed which did not use a state-space description. Perry and Roberts presented a simplified approach based on LC ladder networks and signal flow graphs for bipolar technology [4]. Drakakis & Payne developed the approach using Bernoulli cell building blocks [SI and Fox et al. the E- and E+ building blocks [ 6 ] .Wu and El-Masry [7] recently refined the signal flow method of Peny and Roberts and applied it to MOS transistor circuits. A more general approach was offered by Mulder et al. [SI, who give a general technique for finding the equations of dynamic translinear circuits. The first author has also published a synthesis technique related to log-domain filtering, called the pseudo-voltage technique [9, IO], and has applied this to the design of a silicon cochlea, i.e., a chip that simulates the filtering of sound in the human inner-ear [ I I]. However, the technique proposed here will lead to much more compact filters than the pseudo-voltage technique. Interesting log-domain designs have also been proposed without using any of these synthesis techniques. For example, bipolar and MOS transistors were combined by Punzenberger and Enz lo produce a BiCMOS translinear filter [12]. Python and Enz also designed a Class-AB CMOS log-domain filter for DECT applications [13], proving the viability of CMOS log-domain filters for communications applications. The reason many logdomain designs still occur in an ad-hoc fashion - indeed the main reason there are so many synthesis techniques in the first place - is that none of these techniques seem intuitive and easy to understand or apply. They can he of assistance writing down the equations of a particular logdomain filter, but are not instructive in actually building the circuit. Few designs have been published based on these synthesis methods, even by the creators of the methods.

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In this paper we propose a method, the tau-cell method, that has similarities to the state-variable methods of filter realization, which has been around for several decades for the realization of voltage domain filters [I41 and in the area of systems control, and can be applied. directly to current domain filters. In contrast to the standard frequency domain realization of filters, the state-variable methods are not limited to the implementation of linear filters only. A state-variable method uses feedback to the input of a cascade of first order elements (integrators or leaky integrators) from intermediate points in the cascade and a combination of the intermediate terms in the cascade to create the final output signal as shown in Figure 1. (*)bo

(*)bi

In contrast, in the tau-cell technique all feedback is local, i.e., feedback is only given to the previous tau-cell in the cascade, whereas in the state-variable method each intermediate term feeds back to the input of the cascade. A major disadvantage of the state-variable method is that the feedback terms are all different order functions of certain filter parameters. which makes higher order equations hard to realize 1141. The choice of feedback is much simpler in the tau-cell method, which is expected to yield circuits that are easier to control.

A

A

A

'

Ai

0

Figure 3. A tau-cell cascade.

2. THE TAU-CELL PARADIGM The hasic element in this approach is called a tau-cell. The tau-cell has two programmable parameters: (i) a time constant , T,,and (ii) a current feedback gain, Ai. One of many possible implementations of the tau-cell is shown in Figure 2. It is constructed around a translinear loop consisting of transistors M I , M2, M,, and .&I The loop also contains a capacitor C , two current sources Io used for the readout, the variable current gain A, introducing Ailu into the loop and a feedback current source which introduces A,loij+l/ii, where is the output current of the next tau-cell in the cascade, as shown in Figure 3 . The relation between ii., , ii , and ii,, for this cell is given as: .

-+ E. Figure 1. A state variable method to realize a given transfer function.

,

Ai

i,.,

=

[ris+l-Ai] ii + Ai ii,,

(1)

The time constant of the tau-cell, xi, is given by CUT&,, where UT is the thermal voltage (kT/q). From this it follows that locan he used to program the time constant q of each individual cell. A cascade of tau-cells is shown in Figure 3. Using (I), it can be shown that the transfer function of each tau-cell in the cascade is given by:

and is thus a function of the transfer function of the next tau-cell in the cascade. This is a result of the feedback term Ai iail in ( I ) . Altematively, we can write the transfer function as a ratio of two polynomials in s: Ti =

El= Di -

1 [~is+l-Ai]+ AiNi+,/Di+,

Di+i [qs+l-Ai]Di+,+ AiNj+,

(3)

From this it follows that N, = DcI and thus that:

Di = with

Figure 2. A possible implementation of a tau-cell

[ris+l-Ai] D,+IC A , Di+2 Dk+l= 1 and Dk= T ~ + S 1,

(4)

which recursively defines Di. The transfer function of each block can now be written as:

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(5)

The transfer function from the input of the cascade to the output of block i is the product of the transfer functions of the individual tau-cells and can thus he written as:

0

AI

(a) As Dj is a polynomial in s of order ski1-'.the denominator of every Hi, i.e., D I , is of the order sk and the maximum order of the numerator sk-' is obtained for H I . (Of course the input signal itself can he written as ii, = ij. DJD,, which gives us a numerator of order s'.) Combining the output currents of each section with appropriate weights now allows us to create a system which solves X(s) io, = Y(s) ii,,, where Y ( s ) is a polynomial in s of order k and X(s) can he a polynomial in s up to order k. This way, the cascade of tau-cells can solve arbitrary linear differential equations and a chip containing several of these cascades can he turned into an analogue computer. A

1 m 1 Figure 4. A possible circuit to implement the feedback term. The feedback term AjIoii+l/iican be implemented with another translinear circuit. such as the one shown in Figure 4. If transistors MI-M4operate in weak inversion, the translinear principle states that i,,, ij = Ajloij+lso that io",= AiI0i5+,/ij.

0 (b)

Figure 5 (a) Basic fin1 and (b) second order filler stmctures using tau-cells. Figure Sa shows the first order filter structure with the tau-cell. The feedback term A=O, which retums the taucell to a normal first-order low-pass filter: i, / ii. = 1 / [ T ~ S + I]. Simply taking ii. - i, as the output current yields a first order high-pass filter: iout = T ~ /S [TIS t I]. We have already used three of these first order low pass filters to model a set of three coupled differential equations describing the Inner Hair Cell [Is]. Figure 5h shows the second order filter structure. There are three independent parameters, T ~ T, ~ ,and A , , which together determine the filter time constant and quality factor T and Q. There exists therefore a range of possible combinations that will yield the same theoretical filter characteristics, but which will have different physical implementations. Three interesting possible combinations should he noted. The first case is when T , = x2 = 7. i.e., both tau-cells have the same time constant, which is also the time constant of the filter. A I then has to he chosen as 2 - l/Q in order to obtain a proper second order low-pass filter with:

(7)

3. FIRST AND SECOND ORDER FILTERS

Other filters can be made from this structure using combinations of i,,,, i,, and i2 as the final output of the filter: a high-pass filter is obtained for i,,, = i,, ( i l - iz)/Q - il, a hand-pass filter for io", = ( i l - i2)/Q. and a band-stop filter for iouf = ii, - (i, i2)/Q.

It is well known that a linear filter of any order can he broken down into individual second-order components and one first order component if the overall filter order is odd. This also opens the way for a simpler implementation of tau-cell-based filters. as we can create these out of first and second order sections.

The second case is when T = rl/Q = Q T ~In. this case the feedback term A , = Q', which is a slightly more complex feedback than in the first case. However, creating the different filters now simplifies to: low-pass for io,, = i2, high-pass for i,,, = i,,, i,, band-pass for io", = i, il, and hand-stop for io", = i,. - i , + i2.

~

~

~

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~

Circuits and Signal Processing, vol: 22, pp. 127146,2000. R. Fox, M. Nagarajan. and J. Harris, "Practical Design of Single-Ended Log-Domain Filter Circuits," Proceedings c/l$CAS Hong Kong, vol. I . pp. 341-344. 1997. Wu and El-Masry, "Log-domain synthesis of nthorder filter." International Journal on Electronics. vol. 84, pp. 359-369, 1998 I. Mulder, W. A. Serdijn, A. C. van der Woerd, and A.' H. M. van Roermund, Dynamic Translinear and Log-Domain Circuits: Analysis and Synthesis. Dordrecht: Kluwer Academic Publishers. 1999. E. Fragnikre, E. Vittoz. and A. van Schaik. "A logdomain CMOS transcapacitor: design, analysis and applications." Analog Integrated Circuits & Signal Processing, vol. 22, pp. 195-208.2000. E. Fragniere. A. van Schaik. and E. Vittoz, "Reactive components for pseudo-resistive networks," Electronics Letters, vol. 33, pp. 191314, 1997. A. van Schaik and E. Fragniere, "Pseudo-VoltageDomain Implementation of a Design of a 2Dimensional Silicon Cochlea," presented at Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS 2001). 2001. M. Punzenherger and C. C. Enz. "A Compact LowPower Log-Domain Filter," IEEE Journal of SolidState Circuits, vol. 32. 1998. D. Python and C. C. Enz, "A Mircopower Class-AB CMOS Log-Domain Filter for DECT Applications," IEEE Journal of Solid-State Circuits, vol. 36, 2001. K. L. Su, Analog Filters: Chapman&Hall. 1996. A. McEwan and A. van Schaik, "A Silicon Representation of the Meddis Inner Hair Cell Model,' presented at Proceedings of the ICSC Symposia on Intelligent Systems & Application (ISA'2000). Canada, 2000.

The third case has the simplest feedback term A , = I , for 5

= QTI = TJQ. In this case, however, creating the

different types of filters involves dividing by Q': low-pass . . for iout = 12, high-pass for io", = ii, - (il - i2)/Q2- i2, a band-pass filter for io", = (i, - iz)/Q2.and a hand-stop filter for io., = ii, - (i, il)/Q2. ~

4. CONCLUSIONS In this paper we presented the tau-cell, a building block based on a log-domain low-pass filter, which allows the creation of arbitrary differential equations by placing several tau-cells in a cascade. With a tau-cell implementation all feedback is local, i.e., each first-order section only feeds back to the previous first-order section in the cascade. For a given linear transfer function Hi we can determine possible combinations of time constants, T,, and feedback gains, Ai, using the recursively defined polynomial Di (see equation (4)). As an example, first and second order filters created with the tau-cell approach were discussed.

5. REFERENCES Y. Tsividis, "Continuous-Time Filters in Telecommunications Chips," IEEE Communications Magazine, 2001 D. R. Frey, "Log-domain Filtering: an approach to current mode filtering," IEE Proceedings G, vol. 140, 1993. C. Tomazou and F. J. Lidgey, "Universal active filters using current conveyors." Electronics Letters, vol. 22, 1986.

D. Perry and G. W. Roberts, "The Design of Logdomain Filters Based on the Operational Simulation of LC Ladder Networks," IEEE Transactions on Circuits and Systems 11: Analog andDigital Signal Processing vol. 43. 1996. E. M. Drakakis and A. 1. Payne, "A Bemoulli CellBased Investigation of the non-Linear Dynamics in Log-Domain Structures," Analog Integrated

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