1 / 24

Lecture 16 Random Signals and Noise (III)

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.

nevan
Download Presentation

Lecture 16 Random Signals and Noise (III)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 16 Random Signals and Noise (III) Fall 2008 NCTU EE Tzu-Hsien Sang 1 1 1 1

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

  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., my(t), Ry(), … • Assume X(t) stationary (or WSS at least) H() is LTI. 3 3

  4. 4 4

  5. 5 5

  6. Gaussian Random Process: X(t) has joint Gaussian pdf (of all orders). • 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). 6 6

  7. 7 7

  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(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 ). 8

  9. Example: RC filter with white Gaussian input. 9

  10. 10

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

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

  13. 13

  14. 14

  15. 15

  16. 16

  17. 17

  18. 18

  19. 19

  20. 20

  21. 21

  22. 22

  23. Example: A bandpass signal 23

  24. 24

More Related