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)

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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)

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Quantization Distorsion 2(2)

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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

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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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