IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 51, NO. 11, NOVEMBER 2004 2235 Modular Log-Domain Filters Realized Using Wave Port Terminators Nikos Fragoulis, Member, IEEE, Costas Psychalinos, Member, IEEE, and Ioannis Haritantis Abstract—Log-domain wave filters that simulate the passive LC ladder prototype filters are introduced in this paper. The proposed circuits are constructed from the wave equivalents of the reactive elements corresponding to those in the prototype circuit. The wave equivalent circuits are obtained by using a novel log-domain wave port terminator. The wave equivalent of a capacitor in a shunt branch was chosen as an elementary building block for creating high-order filters. The wave equivalent circuits of all other reactive elements in a shunt or in a series-branch connection can be readily obtained using the elementary building block plus some inverters. This way the derived high-order filter configurations are modular. A design example of a third-order elliptic low-pass filter is given, and the performance of the filter was verified by simulation. Index Terms—Analog integrated circuits, analog wave filters, log-domain filters. I. INTRODUCTION R ECENTLY, there was a significant research effort towards the design of log-domain circuits. This is due to their com- panding nature, as the input signal is initially compressed by an appropriate circuitry and then processed by the core log-domain block. In order to maintain the linear operation of the whole system, the output signal is expanded by a suitable configura- tion. This way, operation at a low-voltage environment can be achieved in log-domain circuits. In addition, log-domain filters are suitable for high-frequency operation, and can be tuned by using dc current sources [1]. A number of methods for designing log-domain circuits have been reported in the literature. These can be classified as follows: state-space (SS) and signal flow graph (SFG) design methods [1]–[3]. According to the SS method, the equations of the prototype system are mapped into the corresponding log-domain equations. An alternative method that offers sim- plicity in the design procedure is introduced in [4]. This is based on the operational simulation of doubly terminated LC ladder prototypes, using the well-known SFG approach and a set of complementary operators. In order to implement the derived log-domain SFGs, lossless and lossy integrators are required. Manuscript received December 13, 2003; revised April 9, 2004. This paper was recommended by Associate Editor Y. Lian. N. Fragoulis was with the Electronics Laboratory, Physics Department, Uni- versity of Patras, Patras GR-26500, Greece. He is now with Analog VLSI Lab- oratory, Ellemedia Technologies, Patras, Greece (e-mail: haritant@physics.up- atras.gr). I. Haritantis is with the the Electronics Laboratory, Physics Department, University of Patras, Patras GR-26500, Greece (e-mail: haritant@physics.up- atras.gr). C. Psychalinos is with the Aristotle University of Thessaloniki, Physics Department, Electronics Laboratory, GR-54124, Thessaloniki, Greece (e-mail: cpsychal@physics.auth.gr). Digital Object Identifier 10.1109/TCSI.2004.836837 Another method for designing high-order filters that simu- late the topology of the corresponding LC ladder prototypes is the wave method [5]–[9]. According to this approach, the corresponding LC passive prototype filter is split into two-port subnetworks that are considered resistively terminated at both ports. Each port is fully described by using the wave variables, defined as incident and reflected waves. Accordingly, these two-port subnetworks are described by using the scattering parameters [7]–[9]. However, the wave variables of each port can be more easily obtained by a circuit adopted at each port, which is the wave port terminator (WPT) [5], [6]. Assuming equal resistive termination at each port, the interconnection be- tween various wave subnetworks is achieved by a cross-cascade connection, that is, the incident wave at each port is equal to the reflected wave of the adjacent port, regardless of the approach used for obtaining the wave subnetworks. Circuits derived by following both of these approaches are known as wave active filters (WAFs). The key point for designing WAFs is the derivation of elemen- tary two-port blocks, according to the wave description of each oneofthetwo-portsubnetworks.Theseblocks,which,aswemen- tioned, are named wave subnetworks, are actually relying on lossy integrator configurations either in the linear or in the log domain. Consequently, the essential difference between the leapfrog and the wave method is that leapfrog-type circuits are relying on loss- less integrators, while wave filters are relying only on lossy inte- grators. Due to the fact that absolute losslessness is not possible in integrated circuit technology, the implementation of wave cir- cuits in integrated form appears to be more realistic. In this paper, a novel log-domain WPT circuit is proposed. The derivation of the proposed terminator became possible after an explicit description of the wave variables in the log domain by using properly chosen operators. Accordingly, the log-do- main WPT is used for obtaining the wave equivalent circuit of a capacitor in shunt branch, which is chosen as an elementary building block upon which the design of the whole filter is relying. The wave equivalents of or reactive elements, in shunt or in series branches, can be derived according to this elementary block and the proper use of inverters. As a result, modular high-order wave log-domain filters can be derived. In Section II the proposed log-domain WPT configuration is presented. Using this block, the wave equivalents of all other reactive elements are obtained. A study of the effects of the bipolar junction transistor (BJT) imperfections is presented in Section III. In Section IV, a third-order elliptic low-pass filter was designed. Using HSPICE simulator, the large-signal frequency responses and the nonlinear behavior of the filter configuration were obtained. 1057-7122/04$20.00 © 2004 IEEE