Lecture 16 Random Signals and Noise (III)

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Lecture 16 Random Signals and Noise (III). Fall 2008 NCTU EE Tzu-Hsien Sang. 1. 1. 1. 1. Outline. Terminology of Random Processes Correlation and Power Spectral Density Linear Systems and Random Processes Narrowband Noise. Linear Systems and Random Processes.

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### Lecture 16 Random Signals and Noise (III)

Fall 2008

NCTU EE

Tzu-Hsien Sang

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Outline
• Terminology of Random Processes
• Correlation and Power Spectral Density
• Linear Systems and Random Processes
• Narrowband Noise
Linear Systems and Random Processes
• Without memory: a random variable  a random variable
• With memory: correlated outputs
• Now, we study the statistics between inputs and outputs, e.g., my(t), Ry(), …
• Assume X(t) stationary (or WSS at least) H() is LTI.

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• Special Case 1: Stationary Gaussian random process

Mean = mx; auto-correlation = RX().

• Special Case 2: White Gaussian random process

RX(t1,t2) = (t1-t2) = () andSX(f) = 1 (constant).

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Properties of Gaussian Processes

(1) X(t) Gaussian, H() stable, linear Y(t) Gaussian.

(2) X(t) Gaussian and WSS X(t) is SSS.

(3) Samples of a Gaussian process, X(t1), X(t2), …, are uncorrelated  They are independent.

(4) Samples of a Gaussian process, X(t1), X(t2), …, have a joint Gaussian pdf specified completely by the set of means and auto-covariance function .

• Remarks: Why do we use Gaussian model?
• Easy to analyze.
• Central Limit Theorem: Many “independent” events combined together become a Gaussian random variable ( random process ).

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Noise equivalent Bandwidth

It is just a way to describe a band-limited noise with the bandwidth of an ideal band-pass filter.

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Narrowband Noise
• Q: Besides certain statistics, is there a more “waveform-oriented” approach to describe a noise (or random signal)?

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