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TSDT14 Signal TheoryLecture 7
Price’s Theorem, Saturation and Quantization
Mikael Olofsson
Department of EE (ISY)
Div. of Communication Systems
2014-09-18 TSDT14 Signal Theory - Lecture 6 2
Price’s Theorem
Useful to determine the ACF after a nonlinear mapping of Gaussian processes.
Prerequisites: (A,B ) is jointly Gaussian with mean (0,0),f (a ) and g (b ) are functions, usually nonlinear,and ρ = E{AB }.
Then we have:
2014-09-18 TSDT14 Signal Theory - Lecture 6 3
Example Price’s Theorem 1(2)
2014-09-18 TSDT14 Signal Theory - Lecture 6 4
Example Price’s Theorem 2(2)
2014-09-18 TSDT14 Signal Theory - Lecture 6 5
More Non-Linearities
From Tables & Formulas, Page 15.
2014-09-18 TSDT14 Signal Theory - Lecture 6 6
Half-Wave Rectifier
From the table:
Complete Maclaurin expansion:
is double factorial (product of every second positive integer):
2013-09-23 TSDT14 Signal Theory - Lecture 7 7
Quantization Principles
2013-09-23 TSDT14 Signal Theory - Lecture 7 8
Uniform Quantization
2013-09-23 TSDT14 Signal Theory - Lecture 7 9
Quantization Distorsion 1(2)
2013-09-23 TSDT14 Signal Theory - Lecture 7 10
Quantization Distorsion 2(2)
2013-09-23 TSDT14 Signal Theory - Lecture 7 11
SDR for Uniform Quantization
Example: Uniform distribution over [-A,A].
Still limited to [-A,A] and nice enough distribution.
⇒
2013-09-23 TSDT14 Signal Theory - Lecture 7 12
SDR for Uniform Quantization with Saturation
Q and S uncorrelated:
Uniform distribution over [-B,B].
2013-09-23 TSDT14 Signal Theory - Lecture 7 13
Non-Uniform Quantization
2013-09-23 TSDT14 Signal Theory - Lecture 7 14
SDR for Non-Uniform Quantization
Uniform Quantization Non-Uniform Quantization
2013-09-23 TSDT14 Signal Theory - Lecture 7 15
Modelling Quantization of a Stochastic Process
2013-09-23 TSDT14 Signal Theory - Lecture 7 16
The Quantization Noise is Almost White 1(4)
Quantization Noise:
Model of PSD:
Assumptions:
Objective:
Show
2013-09-23 TSDT14 Signal Theory - Lecture 7 17
The Quantization Noise is Almost White 2(4)
ACF of the Quantization Noise for k ≠ 0:
2-D PDF of the Quantization Noise:
2-D Taylor series expansion of gives us:
2013-09-23 TSDT14 Signal Theory - Lecture 7 18
The Quantization Noise is Almost White 3(4)
What about all those coefficients? Symmetry:⇒
Result:
Observation:
Result:
2013-09-23 TSDT14 Signal Theory - Lecture 7 19
The Quantization Noise is Almost White 4(4)
We had:
Upper bound:
⇒
Result:
Almost white. Closer to white as ∆ decreases.
Conclusion:
2013-09-23 TSDT14 Signal Theory - Lecture 7 20
Input & Quantization Noise Almost Uncorrelated
Same assumptions:
Normalized cross-covariance:
Objective:
Show
2013-09-23 TSDT14 Signal Theory - Lecture 7 21
Modelling Quantization Noise
Observation:
⇒ & even ⇒
⇒
Almost uncorrelated. Less correlated as ∆ decreases.
Conclusion:
Similar reasoning as before:
⇒
2013-09-23 TSDT14 Signal Theory - Lecture 7 22
Quantization – ACF & PSD Relations
Assumptions:
ACF of output:
Uncorrelated processes:
Result:
2013-09-23 TSDT14 Signal Theory - Lecture 7 23
Quantization – Power-Spectral Densities
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