EECS 242: Volterra/Wiener Representation of Non-Linear...
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EECS 242: Volterra/Wiener Representation
of Non-Linear Systems
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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
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Non-Linear Order-N Convolution
Consider a degree-n system:
kernel
If
Change of variables -
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Generalized Convolution
Generalization of convolution integral of order n:
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Non-Linear Example
x(t) y(t)
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Non-Linear Example (cont)
Kernel is not in unique. We can define a unique “symmetric” kernel.
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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
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Symmetric kernel
We create a symmetric kernel by
System output identical to original unsymmetrical kernel
Volterra Series: “Polynomial” of degree N
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Volterra Series
power series: we get ordinary
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.
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Non-rigorous “proof”
Say y(t) is a non-linear function of for all (all past input)
Fix time t and say that can be characterized by the set so that y(t) is some non-linear function:
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Non-Rigorous Proof (cont)
Let be an orthonormal basis for the space
Thus
“inner product”
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Non-Rigorous Proof (cont)
Expand f into a Taylor series
This is the Volterra/Wiener representation for a non-linear system
Sifting Property:
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Interconnection of Non-Linear Systems
Sum: y
x
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Product Interconnection
y x Product:
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Volterra Series Laplace Domain
Transform domain input/output representation Linear systems in time domain
Define Generalized Laplace Transform:
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Volterra Series Example Generalized transform of a function of two
variables:
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Properties of Transform
Property 1: L is linear Property 2:
Property 3: Convolution form #1
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Properties of Generalized Transform
Property 4: Convolution Form #2:
Property 5: Time delay
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Cascade #1:
Cascade #2:
Cascades of Systems
non-linear linear
non-linear linear
x y
x y
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Cascade Example
x y
property #1 property #2
not symmetric
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Exp Response of n-th Order System
continued
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Exponential Response (cont)
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The Final Result…
We’ve seen this before… A particular frequency mix has response
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Frequency mix response
Sum over all vectors such that
If is symmetric, then we can group the terms as before
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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”
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Example 1
Excite system with two-tones:
x y v
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Example 1 (cont)
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Example 2 Non-linear system in parallel with linear
system:
x y
y1
y2
linear
non-linear
composite
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Example 2 (cont)
assuming H2 is symmetric
Notation:
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Example 2 Again
Redo example with growing exponential method
Overall system is third order, so apply sum of 3 exponentials to system
x y
y1
y2
linear
non-linear
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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
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Example 3 (cont)
Now the product of & produces terms like
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Capacitive non-linearity Non-linear capacitors:
Small signal (incremental) capacitance
BJT
MOSFET
Cj
Vj
small signal cap cap/V
cap/V2
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Cap Non-Linearity (cont)
Model:
Non-Linear Linear
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Overall Model
+
v
-
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Cap Model Decomposition
Let
.
.
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A Real Circuit Example
Find distortion in vo for sinusoidal steady state response
Need to also find
(Note: DC Bias not shown)
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Setup non-linearities Diode:
Capacitor:
Circuit Example (cont)
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Second-Order Terms
Solve for A and B
(1)
(2)
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Third-Order Terms
Solve for A3 & B3
(1)
(2)
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Distortion Calc at High Freq
Compute IM3 at 2ω2-ω1 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:
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At low frequency:
Disto Calc at High Freq (2)
Conclude that at high frequency all third order distortion (fractional) ∝ (signal level)2 for small distortion all second order ∝ (signal level)
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Similarly
Low Freq:
No fixed relation between HD3 and IM3
harmonics filtered and reduced substantially
Disto Calc at High Freq (3)
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Let
Look for
High Freq Distortion & Feedback +
˗
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First-order: Second-order: Frequency dependent
loop gain
High Freq Disto & FB (2)
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Feedback reduces distortion at low frequency and high frequency for a fixed output signal level
True at high frequency if we use where ω 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
Comments about HF/LF Disto
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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
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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.