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       The Algorithm for Multi-tone Response Evaluation of Nonlinear Systems           


The Algorithm for Multi-tone Response Evaluation of Nonlinear Systems

During multi-tone frequency domain analysis, it is assumed that IMD and harmonic measurements for each individual nonlinear element are obtained when the input and output ports have 50Ω terminations and the applied RF sources represent a single-tone bandpass input.

For example, the measurements obtained for a two-port nonlinear amplifier typically represent the AM-AM and AM-PM distortion (e.g., output power and phase vs. input power). These measurements are typically obtained for a given input frequency (or set of frequencies) while the available input power is swept. These measurements are then used during multi-tone frequency domain analysis (taking the operating point for the nonlinear element into account) for computing the IMD generated by this element when it is embedded in a general topology like the one shown in Figure 1 earlier.

Another example is measuring harmonics generated by a mixer when the RF and IF ports are both terminated in 50Ω and a single RF tone (with a given frequency and available input power) is applied to the mixer’s RF port. The measured harmonics (known as MIXERSPURS data) may then be used to predict the harmonics generated by this element when it is embedded in a general topology like the one shown in Figure 1.

The nonlinear frequency domain algorithm described here accounts for all non-linearities and inter-stage mismatches in the system.

For a multi-channel nonlinear topology with N nonlinear elements (refer to Figure 1), it is always assumed that parallel nonlinear channels connected to the same linear multi-port subsystem are not coupled (i.e., non-interacting). This is ensured during multi-tone analysis by assuming that for all nonlinear elements. This effectively implies that the topology has a feed-forward nature. As a result of that, the impedances Zinn and Zoutn for the nth nonlinear element () are only a function of S11 and S22 respectively for the nth nonlinear element () (i.e., each nonlinear element is effectively unidirectional). With that important assumption, the algorithm used for evaluating the IMD and harmonic responses, within the system, proceeds as follows:

1. The topology is partitioned into linear electrical subsystems and nonlinear elements and all the partitioned components are scheduled from the left (input) to the right (output).

2. Harmonics and IMD products at the inputs of each linear electrical subsystem are transformed linearly. The internal and external harmonics and IMD voltage responses may be computed using KCL linear equations at each harmonic or IMD frequency. This calculation takes into account all impedance mismatches Zinn, Zoutn seen looking into nonlinear elements connected to each linear subsystem (refer to Figure 1).

3. Harmonics at the input of each nonlinear element are transformed based on the nonlinear transformation characteristics of that element. This calculation would take into account all impedance mismatches Zsn, Zln seen looking into the ports of other elements connected to each nonlinear element (refer to Figure 1). These impedances are evaluated at all input and output harmonics. Calculations of harmonics and IMD products generated by mixers and nonlinear amplifiers will be discussed later.

The multi-tone frequency domain analysis results may be viewed in the spectral or time domain.

When two or more carriers are applied to a system that includes one or more nonlinear components, inter-modulation frequency components will be generated. The multi-tone domain analysis predicts these inter-modulation distortions (IMD) at all external output ports and at internal nodes where voltage and/or power probes are placed.

The multi-tone frequency domain analysis accounts for all the inter-modulation frequency components generated by all the nonlinear elements within a system. This analysis tends to be more accurate if nonlinear measurements (e.g., Pout vs. Pin) are provided for nonlinear electrical components (as opposed to just providing the nonlinear figures of merit OIP3, P1dB, or Psat).




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