# Modeling Impedance Mismatch with High-Frequency System Simulators

The inability to model impedance-mismatch effects, including the effects of component VSWR and reverse transfer, has always been a major limitation of communication-system simulators. Such effects introduce passband ripple and intersymbol interference, which degrade bit-error rates.

In this article, we will describe a method for including these effects in a system simulator. We will also demonstrate how interface imperfections affect communication system performance.

**Traditional Simulation Approaches**

Most communication system simulators employ highly idealized representations of a system's blocks. System blocks are usually treated as matched, unilateral elements, and passbands are modeled by filters having ideal transmission characteristics. Nonlinear elements are modeled by various means, but the most common is by their memoryless AM-to-AM and AM-to-PM characteristics.

Several evolving trends make these idealizations uncomfortably limiting. The first and most obvious is that such simulators cannot determine the effects of gain and group-delay ripple introduced by imperfect input and output VSWR. Similarly, reverse transfer, which exists in all real components, can create surprisingly large gain variations. These phenomena can have a substantial effect on the bit-error rate in a digital system.

A second important trend is the movement toward top-down design, a process in which the system and its circuit blocks are designed concurrently. In a top-down design methodology, the system is first designed in a system simulator, and circuit specifications are produced. The circuit blocks are then designed, and as the designs are completed, the blocks' simulated performance is entered into the system simulator.

Throughout this process, system simulations are made repeatedly to adequately confirm the block design function in the finished system. Finally, as the circuits are tested, the simulated results are replaced in the system simulator by the test results.

When properly executed, this process allows system designers to remain continuously aware of the effects of circuit performance on system performance. Clearly, however, the process is effective only if the representations of the circuit blocks in the system simulator include all the circuit properties that affect system performance. Thus, accounting for mismatch effects is critical to successful top-down design.

Top-down design inevitably diminishes the gulf between the capabilities of circuit and system simulators. For successful top-down design, the system simulator must adopt many of the characteristics of a circuit simulator. Still, the system simulator cannot be replaced by a circuit simulator; treating a large system as a single circuit is beyond the capability of even the most advanced circuit simulators. Thus, the system simulator must depend on a variety of behavioral methods for modeling its blocks.

**Linear Blocks**

In linear systems, any type of hybrid matrix (for example, impedance [Z], admittance [Y], or scattering [S] parameters) is theoretically adequate for modeling multiport linear networks. Such parameter sets can represent behavioral models for a system simulator as well as circuit elements in a circuit simulator. Given an adequate sample of frequency-domain parameters over an adequate frequency range, they describe linear circuits completely.

RF and microwave circuits are traditionally described by S parameters, so the S-parameter description is a logical one for a communication-system simulator. The approach described in this paper, therefore, uses S parameters for block characterization. Due to conversion relations, however, any n-port hybrid matrix theoretically can be used.

Occasionally it has been observed that frequency-domain models can be noncausal. We believe that noncausality in frequency-domain models is a practical problem, not a theoretical inevitability, and that it can be corrected by proper formulation and characterization of the model and attention to the manner in which Fourier transforms introduce artifacts.

**Nonlinear Blocks**

Modeling nonlinear blocks is a more complex problem and is therefore the focus of much research activity. In general, nonlinear blocks cannot be modeled by hybrid matrices, although hybrid matrices may be useful for such parts as input and output matching circuits. Because of the wide variety of such circuits, it is difficult to create a generalized model for a nonlinear component.

An adequate nonlinear model for many purposes consists of an AM-to-PM block followed by an AM-to-AM block, with input and output filters. The filters have arbitrary S matrices. Reverse transfer is often insignificant in nonlinear components; when significant, it can often be modeled adequately by a linear path.

The characterization of nonlinear blocks can sometimes be simplified by the nature of the circuit that the block represents. Certain kinds of circuits, such as FET power amplifiers, are relatively insensitive to input level. For such circuits, a valid *S* _{11} can be defined. *S* _{22} is more of a problem, however, but methods such as a “hot *S* _{22} ” characterization are potentially valuable.^{1,2} Similarly, reverse transfer effects (*S* _{12} ) in active mixers are invariably negligible.

**Simulating Systems**

The algorithm described in this paper has been implemented in a commercially available digital system simulator that is optimized for digital communication systems. This simulator can operate on either the real signal or its complex envelope representation, although the latter is invariably more efficient for most purposes. Block models include modulators, transmitters, receivers, channel models, filters, and similar objects. The simulator makes full use of object-oriented software design, and each block is an object.

An important capability, for our purposes, is the ability of the simulator to propagate properties throughout the simulated system. This is accomplished using a property set, which includes center frequency, sampling rates, bandwidths, and similar information.

The property set is passed from node to node as part of the model initialization process. This capability has been adapted, in the mismatch-modeling algorithm, to pass mismatch and S-parameter information between blocks.

The operation of the simulator is best described by an example consisting of a simple cascade of linear two-ports. Extensions to multiports and to systems having nonlinear elements is then straightforward.

To account for the effects of component input and output VSWR, all S parameters of each component must be available. These are readily measured or calculated and stored as tables of values. The frequency range and sampling intervals must, of course, be adequate for proper characterization of the device. If necessary, tabulated S parameters can be interpolated by the simulator. In fact, if the component has been designed in the IC analysis tool, the system simulator can automatically launch the circuit simulator and calculate needed data.

**Calculating Mismatch Effects**

To determine mismatch effects, it is necessary to calculate the input and output reflection coefficients of each component, which affect the input or output reflection coefficient of any connected element. Specifically, the input reflection coefficient of a component, Γ_{in} , is:

where &Gamma_{L} is the load reflection coefficient at its output. Since its output termination is the input port of another block, Equaltion 1 must apply to it as well, and so on down the cascade to either the output or to a block having *S* _{12} = 0. Similarly, the output reflection coefficient, Γ_{out} , is:

where Γ_{S } is the source reflection coefficient, usually the output port of another block.

The effect on gain can be analyzed by Mason's Rule.^{4} . This results in a block gain correction term, *S* _{21} *c* , given by:

This is clearly a complex, frequency-dependent quantity, introducing both gain and phase variations with frequency.

Calculating these quantities requires two passes through the cascade of system blocks. (For more complex systems, which can have a number of branches, more passes may be required.) Using the parameter-propagation functionality, the process begins at the first stage and calculates Γ_{out} of each successive elements from Equation 2. It then starts at the last element, calculating Γ_{in} from Equation 1 as it traverses the cascade from output to input. Finally, the block corrections are calculated.

The system simulator performs discrete-time simulations. For such simulations, the frequency-dependent complex transfer function *S* _{21} *c* is modeled in the time domain by means of a finite impulse response (FIR) filter. The filter's coefficients are computed from the complex frequency-dependent mismatch loss. The use of an infinite impulse response (IIR) filter or other digital signal processing techniques is also being explored for block gain correction.

**Simulation Results** **Figure 1** shows the effects of mismatch in a systems simulation of an IEEE 802.11a WLAN architecture. The system simulation models the transmit and receive portion of the RF chain as well as the signal generation and baseband receiver. In this particular system, the impedance mismatch effects are localized to the lowpass filter (LPF) following the downconversion. As a performance metric, we simulate the bit error rate (BER) of the system and show the effects that impedance mismatch has on it.

*Click here for Figure 1*

In **Figure 2** , the magnitude and phase response of the LPF following the downconversion is shown. This filter is constructed in a commercially available RF design tool as a cascade of third-order Butterworth filters, each with a passband cutoff frequency of 25 MHz.

The impedance of each Butterworth filter is designed to be 50 ohms at the input and 100 ohms at the output. Figure 2 plots the response with the impedance mismatch support enabled and disabled in the system simulation tool. With the impedance mismatch support disabled, the 3-dB cutoff frequency is narrower at about 21.6 MHz versus 22.8 MHz with the impedance mismatch option enabled. The passband phase response is relatively unaffected with the impedance mismatch option enabled, but the passband gain response is flatter.

The effects of these variations impact the total system performance and manifest themselves in the system's bit error rate (BER). **Figure 3** shows the BER simulation results with the impedance mismatch option. At a BER of 2e-4, the difference between the curves with the impedance mismatch option enabled is about three quarters of a dB. The curve with the option disabled is slighty worse, as would be expected from the results of Figure 2.

**Wrap Up**

Modeling of mismatch effects is essential for the accurate analysis of communication systems and for top-down electronic system design. Such effects have been incorporated into a comercially available system simulator using FIR filter technology and parameter-propagation functionality. This allows the accurate determination of the effects of mismatch ripple and group delay on system performance. This is critical to the accurate determination of BER performance of complex modulation methods.

**References**

- J. Verspecht and P. Van Esch, “Accurately Characterizing Hard Nonlinear Behavior of Microwave Components with the Nonlinear Network Measurement System: Introducing Nonlinear Scattering Functions,” Proceedings of the 5th International Workshop on Integrated Nonlinear Microwave and Millimeterwave Circuits, pp. 17-26, Duisburg, Germany, October 1998.
- J. Verspecht, M. Vanden Bossche, F. Verbeyst, “Characterizing Components Under Large Signal Excitation: Defining Sensible Large Signal S-Parameters,” 49th ARFTG Conference Digest, pp. 109-117, June 1997.
- Applied Wave Research, Inc., 1960 E. Grand Ave., El Segundo, California, USA, 90245.
- G. Gonzalez,
*Microwave Transistor Amplifiers*, Englewood Cliffs, NJ: Prentice-Hall, 1984. - IEEE Std. 802.11a-1999.

**About the Authors** **Stephen A. Maas** is the chief scientist at Applied Wave Research. He holds a Ph.D. in electrical engineering from UCLA and a BSEE and MSEE from the University of Pennsylvania. Steve can be reached at .

**George Chrisikos** is the engineering director of systems development at Applied Wave Research. He holds a Ph.D. in electrical engineering from the University of Southern California and can be reached at .

**Scotty Hudson** is a development engineer at AWR. He holds a BSEE and an MSEE from the University of Nebraska. Scotty can be reached at.

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