Lecture 16 Random Signals and Noise (III) Fall 2008 NCTU EE Tzu-Hsien Sang

2 Outline Terminology of Random Processes Correlation and Power Spectral Density Linear Systems and Random Processes Narrowband Noise

3 3 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., m y (t), R y (  ), … Assume X(t) stationary (or WSS at least) H(  ) is LTI. 3

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6 6 Gaussian Random Process: X(t) has joint Gaussian pdf (of all orders). –Special Case 1: Stationary Gaussian random process Mean = m x ; auto-correlation = R X (  ). –Special Case 2: White Gaussian random process R X (t 1,t 2 ) =  (t 1 -t 2 ) =  (  ) and S X (f) = 1 (constant). 6

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8 8 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(t 1 ), X(t 2 ), …, are uncorrelated  They are independent. (4) Samples of a Gaussian process, X(t 1 ), X(t 2 ), …, 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 ).

9 9 Example: RC filter with white Gaussian input.

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

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

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23 Example: A bandpass signal

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