EECS 242 Volterra Wiener Representation of Non Linear Systems Linear Input Output Representation A linear system is completely characterized by its impulse response function LTI causality y t has memory since it depends on Non Linear Order N Convolution Consider a degree n system kernel If Change of variables Generalized Convolution Generalization of convolution integral of order n Non Linear Example x t y t Non Linear Example cont Kernel is not in unique We can define a unique symmetric kernel Symmetry of Kernel Kernel h can be expressed as a symmetric function of its arguments Consider output of a system where we permute any number of indices of h For n arguments n permutations Symmetric kernel We create a symmetric kernel by System output identical to original unsymmetrical kernel Volterra Series Polynomial of degree N Volterra Series we get ordinary power series It can be rigorously shown by the Stone Weierstrass theorem that the above polynomial approximates a non linear system to any desired precision if N is made sufficiently large Non rigorous proof Say y t is a non linear function of all past input Fix time t and say that by the set linear function for all can be characterized so that y t is some non Non Rigorous Proof cont Let be an orthonormal basis for the space Thus inner product Non Rigorous Proof cont Expand f into a Taylor series This is the Volterra Wiener representation for a non linear system Sifting Property Interconnection of Non Linear Systems Sum x y Product Interconnection Product x y Volterra Series Laplace Domain Transform domain input output representation Linear systems in time domain Define Generalized Laplace Transform Volterra Series Example Generalized transform of a function of two variables Properties of Transform Property 1 L is linear Property 2 Property 3 Convolution form 1 Properties of Generalized Transform Property 4 Convolution Form 2 Property 5 Time delay Cascades of Systems Cascade 1 x y non linear linear Cascade 2 x y linear non linear Cascade Example x y property 1 not symmetric property 2 Exp Response of n th Order System continued Exponential Response cont The Final Result We ve seen this before A particular frequency mix has response Frequency mix response Sum over all vectors such that If is symmetric then we can group the terms as before Important special case P n To derive we can apply n exponentials to a degree n system and the symmetric transfer function is given by times the coefficient of We call this the Growing Exponential Method Example 1 x v Excite system with two tones y Example 1 cont Example 2 Non linear system in parallel with linear system linear y1 y x y2 non linear composite Example 2 cont assuming H2 is symmetric Notation Example 2 Again linear y1 y x y2 non linear Redo example with growing exponential method Overall system is third order so apply sum of 3 exponentials to system Example 3 We can drop terms that we don t care about We only care about the final term so for now ignore terms except where Focus on terms in y2 first symmetric kernel Example 3 cont Now the product of terms like produces Capacitive non linearity Cj Non linear capacitors BJT MOSFET Vj Small signal incremental capacitance cap V small signal cap cap V2 Cap Non Linearity cont Model Non Linear Linear Overall Model v Cap Model Decomposition Let A Real Circuit Example Note DC Bias not shown Find distortion in vo for sinusoidal steady state response Need to also find Circuit Example cont Setup non linearities Diode Capacitor Second Order Terms 1 2 Solve for A and B Third Order Terms 1 2 Solve for A3 B3 Distortion Calc at High Freq Compute IM3 at 2 2 1 only generated by n 3 3 2 1 1 2 3 H3 is symmetric so we can group all terms producing this frequency mix by H3 For equal amp o p signal we adjust each input amp so that Disto Calc at High Freq 2 At low frequency Conclude that at high frequency all third order distortion fractional signal level 2 for small distortion all second order signal level Disto Calc at High Freq 3 Similarly Low Freq No fixed relation between HD3 and IM3 harmonics filtered and reduced substantially High Freq Distortion Feedback Let Look for High Freq Disto FB 2 First order Second order Frequency dependent loop gain Comments about HF LF Disto Feedback reduces distortion at low frequency and high frequency for a fixed output signal level True at high frequency if we use where is evaluated at the frequency of the distortion product While IM HD no longer related CM TB P 1dB PBL are related since frequencies close together Most circuits 90 can be analyzed with a power series References Nonlinear System Theory The Volterra Wiener Approach Wilson J Rugh Baltimore Johns Hopkins University Press c1981 UCB EECS 242 Class Notes Robert G Meyer Spring 1995 More References Piet Wambacq and Willy M C Sansen Distortion Analysis of Analog Integrated Circuits The International Series in Engineering and Computer Science Hardcover M Schetzen The Volterra and Wiener theories of nonlinear systems New York Wiley 1980 L O Chua and N C Y Frequency domain analysis of nonlinear systems formulation of transfer functions IEE Journal on Electronic Circuits and Systems vol 3 pp 257 269 1979 J J Bussgang L Ehrman and J W Graham Analysis of Nonlinear Systems with Multiple Inputs Proceedings of the IEEE vol 62 pp 1088 1119 1974 J Engberg and T Larsen Noise Theory of Linear and Nonlinear Circuits New Yory John Wiley and Sons 1995