Behavioral Modeling of Transmission Line Channels via Linear Transformations

An approach based on the linear transformation of network port variables is augmented and applied to behavioral modeling of inhomogeneous transmission line channels with lumped and distributed discontinuities. Generic tools Spice-like and Simulink can be used for channel simulation in the frequency or time domain. Introduction Recent developments in very high bit rate wireline data transmission applications as 1000BASE-T Gigabit Ethernet (GE) [1], VDSL [2] and other, have motivated continued researcher interest in accurate simulation of data channels based on the unshielded twisted pair (UTP) medium. As the signaling bandwidths approach the order of 100 MHz, even relatively minor channel discontinuities such as short jumpers, connectors etc. need to be accounted for in the system design. Such discontinuities cause multiple reflections in the channel, becoming an impairment for the system performance. An approach based on the linear transformation of network port variables, originally introduced for synthesis of active filters [3], has been augmented and applied to the behavioral modeling of inhomogeneous transmission line channels with lumped and distributed discontinuities. First, a channel with discontinuities is split into a cascade connection of uniform blocks: two-ports, representing lumped (discrete) circuit elements, or homogeneous sections of transmission line. The bidirectional channel model structure is then derived in a regular procedure, using selectively chosen port variable transformation matrices. The overall model consists of a cascade interconnection of the matrix transfer function modules of the two types. Modules of the first type correspond on an element-by-element basis to the channel discrete instances, or homogeneous transmission line sections, whereas the second type are the so called “port-matching” modules. The later serve as an interface between input and output ports for the adjacent modules of the first type. It is shown that the port-matching requirements result from the discontinuity boundaries in the physical channel and yield a matrix transfer function whose elements are in essence the reflection and refraction coefficients. The overall channel model, referred to as the linear transformation equivalent (LTE), can be readily simulated in time or frequency domain using generic tools like Matlab/Simulink or Spice. Physical phenomena in transmission lines such as skin effect, dielectric loss and characteristic impedance variations can also be modeled as shown. Homogeneous transmission line sections are described in terms of their corresponding characteristic impedance, propagation delay and rational s-domain transfer function approximating insertion loss frequency response. The near and far end echo and transmission waveforms are generated simultaneously. The proposed method offers certain advantages compared to the approach traditionally used in the communications community based on the frequency domain computation of the channel chain matrix ABCD parameters [4], [5], with subsequent frequency-to-time discrete-domain Laplace inversion by IFFT. The LTE model naturally yields a continuous-time system which can be simulated using analog variable-step tools. This provides means for a relatively straightforward and efficient behavioral simulation and analysis of complex mixed-signal systems, especially under conditions of timing reference frequency drift and/or offset, as is the case, for example, with a pair of data communications transceivers with timing recovery. The method also allows effective what-if, or component variation analysis. Application example is given whereby GE timing recovery test channel is modeled together with a typical transformer line interface. Linear Transformation Approach The linear transformation method was originally developed by Dimopoulos and Constantinides [3] with the main aim of active filter synthesis from the doubly terminated lossless LC ladder filters. The philosophy of the approach is that physical variables (currents and voltages) of a network are first linearly transformed into a new domain where the corresponding relationships are then realized by means of transfer functions. By varying the linear transformation parameters different structures corresponding to the same prototype network may be obtained, possessing features desirable for a targeted realization. Due to the universal nature of the underlaying concept, the basic approach could be adopted for applications in system modelling, as demonstrated in this paper for the case of a transmission line channel with distributed and lumped discontinuities. The concept of linear transformations is briefly presented below. Fig. 1. (a) The passive two-port prototype. (b) The SFG of the corresponding LTE. (c) LTE block diagram. For an arbitrary linear passive network (see Fig 1(a)) contained between the ports and with associated voltage-current variable vectors [V1 I1] T and [V2 I2] T, one can assign a two-port N characterized in terms of the modified1 chain matrix ABCD as follows: (1) . The port variable vectors [V1 I1] T and [V2 I2] T are linearly transformed by a pair of nonsingular matrices Λ1 and Λ2 into the new variables [x1 y1] T and [x2 y2] T such that: . (2) where . Fig. 2. (a) The cascade connection of two-ports. (b) The interconnection of the corresponding LTE blocks. Combining (1) and (2) yields: