This document discusses using maximum entropy methods for waveform compensation to overcome limitations in instrument bandwidth. It presents the problem of distorted waveforms due to limited bandwidth and describes using pre-distortion to cancel out the effects of linear distortion. It introduces the maximum entropy method as an algorithm for selecting the best pre-distortion solution from the set of acceptable solutions that minimize the difference between the emulated and target waveforms. The maximum entropy method aims to find a unique solution by maximizing the entropy of the system. Application of this technique is shown to improve rise time and setting time performance beyond the original instrument specifications.

1. AWG waveform compensation By
Maximum Entropy Method
Dr. Fang Xu Teradyne, Inc.
Fang.xu@teradyne.com
st techniques to face new challenges
Developing n ew te
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2. Problem to Solve
Given instrument has limited bandwidth
DIB: Transformer
Actual performance: <80MHz
6nS Rise time and >50nS set to 2% effective
bandwidth
Target performance:
2nS rise time and short setting time
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3. Digital Signal
Processing Can
Make Many Things
Possible
Vico Rota,
Sorrento, Italy
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4. Waveform distortion
Distorted
waveform
Non Equal Create ISI at
amplitude high speed
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5. Consequence
• Difference between the signal produced by
AWG and what we want
• Difficulty of signal level calibrate
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6. Different distortions
• Non linear distortion
• Quantization noise
• Random noise
• Linear distortion
If f(t) and g(t) are complex functions of t, G and H are linear systems,
α and β are two complex numbers, * denotes the convolution product,
Then we have
• H(f(t)) = (H*f)(t)
• H(αf(t)+βg(t)) = α(H*f)(t)+β (H*g)(t)
• ((G*H)(f(t)) = G(H*f)(t)
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7. Pre-Distortion Principle
DAC +
RAM Amplifier Desired
+ etc. output
Pre-distorted Electronic
waveform device
distortion
One cancel the effect
of the other
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8. Matrix Inversion
Target waveform
T = H*G Compensated waveform segment
Linear distortion
A straightforward solution for G should be
G = H-1*T, where H-1 is the
Inverse matrix of H
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9. Inverse matrix limitation
• H can have zeros’. The inverse matrix of H
does not exist.
• The inverse matrix of H does exist, The
response function of the instrument can be a
high loss function for some components. Due
to the capture noise in H and computational
errors, the solution can be very unstable.
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10. General inversion Algorithm
Guess
waveform Captured Emulated
G0’ response H target T0’
Improvement
for G1’ Template
Comparator
C = (T0 '−T )
Step Instrument Target
waveform response waveform T
AWG memory AWG
electronics
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11. Matching Waveform to Target
Make C = (T0 '−T )
2 2
smaller than the limit of noise: Solution is not unique!
Acceptable T’ T0’ = H*G0’
Solutions T
H
G’
Noise level
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12. Need a Second Condition
to select a best solution from all acceptable solutions.
• Maximum likelihood
• Maximum entropy
• others
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13. Entropy: Degree of Disorder
Low level entropy High level entropy
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14. Entropy of a System
A system having n probable events: gn, f n = Fpn
pn is the probability of each event,
fn is the occurrence of event gn, F = ∑ fn
F is the total number of occurrence.
Possible number of combinations:
F!
∏ f n!
Entropy of the system:
S = −∑ pn ln pn
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15. Observation ≡ Entropy Reduction
Any method of observation or measurement reduce
the entropy of a system (like a picture of gas
molecules taken at instant T)
So, The intrinsic entropy of a dynamic system is
always higher than what we observed
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16. Maximum Entropy Method
Maximize a new function:
Acceptable
T’ Q = αS − βC 2
Solutions
T
H
G’ Field of
entropy
Unique
solution
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17. Algorithm and Flow Chart
Gn’ = αlnGn-1’- βHt M (T -H* Gn-1’)+ Gn-1’
Guess of Response Emulated
result Gn’ matrix H target
-
Transpose Optional
matrix Ht weighting
+
Maximizing Desired
entropy target
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18. Pre-Distortion Result
Waveform before compensation
Rise time = 6nS
Setting time >50nS
Compensation waveform
Waveform after Compensation
Rise time = 2.5nS
Preshoot = 0.65%
Overshoot = 0.84%
Undershoot = 0.50%
Setting time = 6.5nS
This technique can also applied to digitizer
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19. Conclusion
• Instruments have linear distortions
• Customers may use the instrument out of its’ specs
• We may want to “extend” the spec of existing
instrument
This method uses Maximum Entropy
Method to correct waveform to let us
generate ideal waveform
We need more and more techniques like this
to overcome instrument physical limitations
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