Modeling Nonlinear Behavior of Band-Pass Memoryless and Dynamic Systems José Carlos Pedro, Nuno Borges Carvalho and Pedro Miguel Lavrador Telecommunications Institute, University of Aveiro – 3810-193 AVEIRO – Portugal Abstract – This paper addresses nonlinear distortion arising in microwave band-pass memoryless and dynamic systems. It first identifies the minimum requirements for their correct representation. Then, it shows that the complex behavior of long term memory effects does not allow a unique characterization procedure, but demands for various nonlinear distortion figures according to the type of nonlinear RF impairments the actual system is sensitive to. I. INTRODUCTION The analysis of band-pass memoryless systems has deserved a strong attention for now more than thirty years [1]. But, recently, the microwave community began to realize that such approximation was insufficient to accurately design wideband power amplifier, PA, linearizers. Consequently, a growing attention has been paid to nonlinear distortion effects arising from band-pass systems showing significant long term memory, and to their accurate modeling [2,3]. The main purpose of this paper is to present the modeling requirements of those microwave band-pass memoryless and dynamic systems, taking into consideration their general nonlinear distortion behavior. Then, those results are used in the discussion of the appropriateness of most important nonlinear distortion figures of merit and their corresponding laboratory measurement set-ups. II. BAND-PASS MEMORYLESS AND DYNAMIC SYSTEMS’ REPRESENTATION Microwave and wireless PAs may present memory effects that have short and long time-constants compared to the RF carrier signal or to its slowly-varying envelope. For the short memory effects contribute the band-pass characteristics of the PA input and output matching networks and, sometimes, also the low-pass characteristics of the active device. These can be modeled by two filters with a memoryless transfer nonlinearity in between. Modeling long term nonlinear memory effects is much more difficult. Theoretical and experimental works have related those effects to a large variety of PA characteristics that span from low frequency dispersion induced by long time-constant traps and thermal constants, deliberate or accidental envelope feedback and long time-constants present in the input and output bias circuitry. Apparently, it seems that, from these, bias circuitry induced memory is, with a more or less extent, common to almost all PA circuits, being particularly important in the output of FET based PAs, and in the input of bipolar transistor ones [2]. In any case, it should be obvious that in usual band-pass microwave PAs intended to handle signals that occupy only a small percentage of their available bandwidth (this way leading to instantaneous responses), there must be some low frequency, LF, component (the envelope), for which the circuit is no longer memoryless, that must be remixed with the original RF signal to create those long term memory effects. But, since these LF envelope components can only be generated (or demodulated from the RF signal) in a nonlinearity, no transfer nonlinearity can remix them again with the original signal to produce new in-band intermodulation products, unless some form of LF feedback is available. Note, however, that this LF feedback needs not to be a physical path, but simply the conceptual feedback present whenever, e.g., the output current of a FET, at the envelope frequencies, generated from the i DS (v GS ) nonlinearity, circulates in the load impedance mesh and is converted into a V ds (ω LF ) that is then remixed with the original V ds (ω RF ) signal in the i DS (v DS ) nonlinearities. Beyond this LF feedback, remixing even order high- frequency, HF, components with the original ones, can also produce in-band intermodulation components. Therefore, memory pressed into these HF components can also be a cause of band-pass dynamic behavior. This is illustrated in the PA simplified circuit of Fig. 1 and the corresponding system model of Fig. 2. i DS v S (t) v DS (t) R 0 v O (t) v GS (t) R 0 Linear Dynamic Matching Network M i (ω) Linear Dynamic Matching Network M o ( ω) i DS = I DS + G m v gs + G ds v ds + G m2 v gs 2 + G md v gs v ds + G d2 v ds 2 + G m3 v gs 3 + G m2d v gs 2 v ds + G md2 v gs v ds 2 + G d3 v ds 3 Z L ( ω) Fig. 1 Simplified FET based PA circuit used for the nonlinear analysis. 0-7803-7695-1/03/$17.00 © 2003 IEEE 2003 IEEE MTT-S Digest IFTH-24 2133