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Figure from:Analysis and compact behavioral modeling of nonlinear distortion in analog communication circuits

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Fig. 4. Nonlinear system described by Volterra operators.

Figure 4 Nonlinear system described by Volterra operators.

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Fig. 1. Design flow for the analog front-ends that links the architectural, circuit, and layout levels of description. Petr Dobrovolny, Member, IEEE, Gerd Vandersteen, Member, IEEE, Piet Wambacq, Member, IEEE, and Stéphane Donnay, Member, IEEE
Fig. 1. Design flow for the analog front-ends that links the architectural, circuit, and layout levels of description. Petr Dobrovolny, Member, IEEE, Gerd Vandersteen, Member, IEEE, Piet Wambacq, Member, IEEE, and Stéphane Donnay, Member, IEEE
Fig. 2. Different steps of the modeling approach in view of various levels of simulation abstraction and domains.
Fig. 2. Different steps of the modeling approach in view of various levels of simulation abstraction and domains.
Fig. 3. Nonlinear equivalent circuit of a bipolar transistor.
Fig. 3. Nonlinear equivalent circuit of a bipolar transistor.
Fig. 5. Simple nonlinear circuit and its equivalent circuit according to a power series expansion of the nonlinear conductance.
Fig. 5. Simple nonlinear circuit and its equivalent circuit according to a power series expansion of the nonlinear conductance.
Evaluating (11) at all possible combinations of frequency ar- guments (w91, w92, 43) from the frequency set 2 completely characterizes the third- order nonlinear behavior of the circuit for a given input excitation. Then, the third-order circuit response at a chosen frequency wou: is determined by Indeed, a practical circuit contains many nonlinear elements and the procedure must be repeated for each of the basic non- linearities in the circuit. Then, the VKTs at the observed output comprise all contributions from all basic nonlinear elements. This implies that (11) can be generalized toward
Evaluating (11) at all possible combinations of frequency ar- guments (w91, w92, 43) from the frequency set 2 completely characterizes the third- order nonlinear behavior of the circuit for a given input excitation. Then, the third-order circuit response at a chosen frequency wou: is determined by Indeed, a practical circuit contains many nonlinear elements and the procedure must be repeated for each of the basic non- linearities in the circuit. Then, the VKTs at the observed output comprise all contributions from all basic nonlinear elements. This implies that (11) can be generalized toward
Fig. 7. Block diagram for the computation of the second-order nonlinear behavior of a nonlinear circuit.
Fig. 7. Block diagram for the computation of the second-order nonlinear behavior of a nonlinear circuit.
Fig. 9. Path from the input to the output of a weakly nonlinear circuit corresponding to the contribution of a 1-D second-order coefficient A°2,1 to the overall third-order nonlinear behavior. Fig. 8. Path from the input to the output of a weakly nonlinear circuit corresponding to the contribution of a 1-D third-order coefficient K'3,; to the overall third-order nonlinear behavior.
Fig. 9. Path from the input to the output of a weakly nonlinear circuit corresponding to the contribution of a 1-D second-order coefficient A°2,1 to the overall third-order nonlinear behavior. Fig. 8. Path from the input to the output of a weakly nonlinear circuit corresponding to the contribution of a 1-D third-order coefficient K'3,; to the overall third-order nonlinear behavior.
Fig. 12. 5-GHz low-noise amplifier. (a) The circuit scheme of the amplifier. (b) The scheme of the transistor subcircuits X -- -. Fig. 11. 5-GHz WLAN receiver f-front-end described at the architecture level (the LNA block high-level specifications are indicated in more detail). The architecture description displayed in Fig. 11 represents a usual starting point for a mixed-signal design flow. The models for this first high-level simulation are typically very rough. For instance, the LNA block is defined as a static nonlinearity, de- scribed by a polynomial of order three that relates output to the input. The coefficients of this polynomial are related to the gain and intercept points [8].
Fig. 12. 5-GHz low-noise amplifier. (a) The circuit scheme of the amplifier. (b) The scheme of the transistor subcircuits X -- -. Fig. 11. 5-GHz WLAN receiver f-front-end described at the architecture level (the LNA block high-level specifications are indicated in more detail). The architecture description displayed in Fig. 11 represents a usual starting point for a mixed-signal design flow. The models for this first high-level simulation are typically very rough. For instance, the LNA block is defined as a static nonlinearity, de- scribed by a polynomial of order three that relates output to the input. The coefficients of this polynomial are related to the gain and intercept points [8].
Fig. 10. Contribution of the 1-D second-order coefficient K2,,, to the third-order nonlinear behavior; (a) bandpass representation (b) complex low-pass equivalent with multicarrier signal representation.
Fig. 10. Contribution of the 1-D second-order coefficient K2,,, to the third-order nonlinear behavior; (a) bandpass representation (b) complex low-pass equivalent with multicarrier signal representation.
“#MIC / #AC means the number of the Most Important Contributions / the number of All Contributions > all nonlinear coefficients are related to the intrinsic NPN transistor Q, of the transistor macromodel X3 afi ee US MostT IMPORTANT CONTRIBUTIONS TO THE 2ND- AND 3RD-ORDER VKT TABLE I
“#MIC / #AC means the number of the Most Important Contributions / the number of All Contributions > all nonlinear coefficients are related to the intrinsic NPN transistor Q, of the transistor macromodel X3 afi ee US MostT IMPORTANT CONTRIBUTIONS TO THE 2ND- AND 3RD-ORDER VKT TABLE I
Fig. 13. Magnitude of the second-order VKT and the important contributions to it around dc and the second harmonic of the input carrier frequency.
Fig. 13. Magnitude of the second-order VKT and the important contributions to it around dc and the second harmonic of the input carrier frequency.
QUANTITATIVE SUMMARY OF DIFFERENT BLOCKS IN THE HIGH-LEVEL MODEL Notice that the model for the third-order behavior comprises also blocks designated H2 — g,, and H2 — g, that repre- sent the second-order V KT of the voltages over the appropriate controlling ports. It illustrates the way in which the second- order nonlinearity coefficients contribute to the third-order be- havior. The high-level model consists only of static nonlinear blocks, transfer function blocks, and scale factors. Structure of the model with respect to the number of different kinds of blocks is recapitulated in Table II.
QUANTITATIVE SUMMARY OF DIFFERENT BLOCKS IN THE HIGH-LEVEL MODEL Notice that the model for the third-order behavior comprises also blocks designated H2 — g,, and H2 — g, that repre- sent the second-order V KT of the voltages over the appropriate controlling ports. It illustrates the way in which the second- order nonlinearity coefficients contribute to the third-order be- havior. The high-level model consists only of static nonlinear blocks, transfer function blocks, and scale factors. Structure of the model with respect to the number of different kinds of blocks is recapitulated in Table II.
Fig. 16. Multicarrier complex low-pass equivalent model of the second-order nonlinear behavior of the LNA. Fig. 15. Comparison of the simulation efficiency with the circuit and the high-level model depending on the number of sinusoidal components covering the frequency band of the input excitation.
Fig. 16. Multicarrier complex low-pass equivalent model of the second-order nonlinear behavior of the LNA. Fig. 15. Comparison of the simulation efficiency with the circuit and the high-level model depending on the number of sinusoidal components covering the frequency band of the input excitation.
Fig. 14. High-level model of the LNA; (a) the second-order nonlinear behavior block (b) the third-order nonlinear behavior block.
Fig. 14. High-level model of the LNA; (a) the second-order nonlinear behavior block (b) the third-order nonlinear behavior block.