IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 1, JANUARY 2005 99 An ON–OFF Log Domain Circuit That Recreates Adaptive Filtering in the Retina Kareem A. Zaghloul and Kwabena A. Boahen Abstract—We introduce a new approach to synthesizing Class fore, to obtain the derivative of the voltage, divide the derivative AB log-domain filters that satisfy dynamic differential-mode of the signal, , by the signal, . That is to say, divide the and common-mode constraints simultaneously. Whereas the current you wish to supply to the capacitor by the current made dynamic differential-mode constraint imposes the desired fil- tering behavior, the dynamic common-mode constraint solves by the transistor whose gate (or base) is connected to it. Intu- the zero-dc-gain problem, a shortcoming of previous approaches. itively, this division compensates for the slope of the exponential Also, we introduce a novel push–pull circuit that serves as a at the transistor’s operating point, such that its current changes current-splitter; it rectifies a differential signal into the ON and at a constant rate. Current-division is readily realized with log- OFF paths in our log-domain filter. As an example, we synthesize a arithmic elements by exploiting the translinear principle [8]. first-order low-pass filter, and, to demonstrate the rejection of dc signals, we implement an adaptive filter by placing this low-pass In theory, log-domain filters have limitless dynamic range; in circuit in a variable-gain negative-feedback path. Feedback gain is practice, dynamic range is limited by the bias current. Seevinck controlled by signal energy, which is extracted simply by summing and Frey have both proposed Class AB log-domain filters complementary ON and OFF signals—dc signals do not contribute that address this shortcoming; they both use two copies of the to the signal energy nor are they amplified by the feedback. We log-domain circuit to filter the differential signal [7], [12]. In implement this adaptive filter design in a silicon chip that draws biological inspiration from visual processing in the mammalian Seevinck’s approach, the outputs are cross coupled, each sub- retina. It may also be useful in other applications that require tracting current from the others capacitor. In Frey’s approach, dynamic time-constant adaptation. a current-splitter, which receives a bidirectional input current, Index Terms—Adaptive filtering, artificial vision, class AB cir- is placed up front; it enforces a geometric mean constraint. cuits, neuromorphic engineering. Unfortunately, both designs suffer from distortion when the filter’s transfer function has zero gain at dc, or close to zero, due to a reduction in bandwidth and to offsets introduced by I. LOG-DOMAIN FILTERING leakage currents. ECREASING supply voltage with integrated circuit In this paper, we introduce a new approach to synthesizing D miniaturization is increasing interest in current-mode Class AB log-domain filters. Our synthesis procedure satisfies filters. Current-mode operation offers large dynamic range if dynamic differential-mode and common-mode constraints the nonlinear device transconductance is compensated for in simultaneously. Whereas the dynamic differential-mode the filter design, such that operation remains linear outside the constraint imposes the desired filtering behavior, as in the small-signal region. The existence of such externally linear approaches of Frey and Seevinck [7], [12], the dynamic but internally nonlinear filters was demonstrated by Adams, common-mode constraint solves the zero-dc-gain problem, a who first designed a circuit that “when placed between a log shortcoming of their approaches. Specifically, we introduce a converter and an anti-log converter will cause the system to second differential equation, with its own time-constant, that act as a linear filter” [1]. He named these circuits log-domain imposes the desired common-mode behavior, and, in particular, filters. The log and anti-log operations are readily realized we find that imposing a geometric mean constraint that is using bipolar transistors or MOSFETs operating in weak in- satisfied with the same time-constant that describes differential version; these devices maintain logarithmic voltage-current behavior results in the simplest implementation. relationships over six decades. The remainder of this paper is organized as follows. In Sec- The principle of log-domain filter design is a simple one: use tion II, we introduce a novel push–pull circuit that serves as a current to represent the signal , voltage to represent its loga- current-splitter in our log-domain filters; it rectifies a differential rithm , and note that . There- signal into ON and OFF paths. In Section III, taking these comple- mentary signals as input, we synthesize a low-pass ON–OFF log- domain filter that constrains the geometric mean of its outputs Manuscript received June 6, 2003; revised July 29, 2004. This work was sup- ported by a National Institutes of Health Vision Training Grant (T32-EY07035) dynamically. In Section IV, taking inspiration from the retina, and by the Whitaker Foundation under Grant 37005-00-00. The work of K. A. we realize an adaptive filter by placing our ON–OFF log-domain Zaghloul was also supported by a Ben Franklin Fellowship from the University low-pass in a variable-gain negative-feedback path. Feedback of Pennsylvania School of Medicine. This paper was recommended by Asso- ciate Editor P. Arena. gain is controlled by signal energy, which is extracted simply by K. A. Zaghloul is with the Department of Neuroscience, University of Penn- summing complementary ON and OFF signals. This application sylvania, Philadelphia, PA 19104 USA. demonstrates the rejection of dc signals—they are not amplified K. A. Boahen is with the Department of Bioengineering, University of Penn- sylvania, Philadelphia, PA 19104 USA. in the feedback path nor do they contribute to the signal energy. Digital Object Identifier 10.1109/TCSI.2004.840097 Section V concludes the paper. 1057-7122/$20.00 © 2005 IEEE 100 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 1, JANUARY 2005 To determine the behavior of these ON–OFF signals subject to this constraint, we observe that (1) implies Replacing the sum of and with , where is the common-mode input signal, we have Fig. 1. Half-wave rectification. (a) Circuit implementation of half-wave (2) rectification. Two currents s and s are compared to one another. Both currents are mirrored on to one another, eliminating most of the common mode (i.e., dc) current and driving subsequent circuitry with the differential Thus, the geometric mean of and is strictly less than signals, s and s . determines the level of residual dc signal present in s and s . Additional copies of s and s can be made to .If , then . Conversely, if drive subsequent circuitry by connecting additional transistors in parallel. (b) , then . Consequently, and In a purely differential representation, the difference between s and s , in the first case, while and s , is encoded as the difference between s and s (top). In the ON–OFF representation, one signal or the other is active, depending on the sign of the in the second case. We can see that the circuit rectifies its in- difference between the input signals (bottom). When s a s , residual dc puts around a level determined by . Hence, once exceeds s currents are inversely proportional to the common-mode input, . Tick by several , current is diverted entirely through the OFF s marks on the horizontal axis of both graphs represent units of . path. Conversely, once exceeds by several , current is diverted entirely through the ON path. Whereas a conven- II. ON–OFF RECTIFICATION tional differential circuit would maintain current in both paths (Fig. 1(b), top), our ON–OFF design maintains current in only one To construct our class AB log-domain filter, we first construct path as shown in the analytical solution presented in the bottom a circuit to divide input signals into complementary ON and OFF of Fig. 1(b). paths. Taking inspiration from Frey’s current-splitter [7], we We can also determine the predicted quiescent level of subtract the two input currents and completely divert the dif- and when , which represents the ference to the ON or OFF path based on its sign. common-mode input current level, from (2) A. Implementation (3) We implement rectification using the circuit shown in Fig. 1(a). This circuit is similar to that proposed in [14], but (4) the analysis presented in that paper includes effects of , which we ignore here. The circuit takes two input signals, and when . Hence, the common-mode rejection in our , on either side, and compares them to one another. In our ON–OFF circuitry is in fact not complete. Its outputs contain application (see Section IV-A), these currents represent a signal a residual dc component that is linearly proportional to and its mean, such that current is diverted to either the ON or and inversely proportional to the common-mode input signal, OFF pathways based on whether the signal lies above or below as shown in Fig. 1(b). its mean. We define a current that sets the Finally, because we have assumed the transistors are in sat- residual current level and assume a unity subthreshold slope uration, our results do not apply to input currents coefficient (i.e., ). Hence, the currents in the current ,or (since mirror can be expressed as and . and ). For currents above this level, the Equating these currents to the input and output currents, we current mirrors’ output transistors enter the ohmic region, and find hence and start leveling off. The maximum level they can achieve is . (1) B. Simulation Results To verify our rectifying ON–OFF design, we simulated the cir- assuming subthreshold operation ( cuit of Fig.