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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing- Continuous and Discrete Fall Term 2008 Lecture 22 1 Reading: Proakis and Manolakis: Secs. 12,1 12.2 Oppenheim, Schafer, and Buck: Stearns and Hush: Ch. 13 1 The Correlation Functions (continued) In Lecture 21 we introduced the auto-correlation and cross-correlation functions as measures of self- and cross-similarity as a function of delay . We continue the discussion here. 1.1 The Autocorrelation Function There are three basic definitions (a) For an infinite duration waveform: T/ 2 1 ff ( ) = lim f ( t ) f ( t + )d t T T T/ 2 which may be considered as a power based definition. (b) For an finite duration waveform: If the waveform exists only in the interval t 1 t t 2 t 2 ff ( ) = f ( t ) f ( t + )d t t 1 which may be considered as a energy based definition. (c) For a periodic waveform: If f ( t ) is periodic with period T t + T 1 ff ( ) = f ( t ) f ( t + )d t T t for an arbitrary t , which again may be considered as a power based definition. 1 copyright c D.Rowell 2008 221 f ( t ) t T a f ( t + J ) t a- J T- J H Example 1 Find the autocorrelation function of the square pulse of amplitude a and duration T as shown below. B J J 6 The wave form has a finite duration, and the autocorrelation function is T ff ( ) = f ( t ) f ( t + )d t The autocorrelation function is developed graphically below T ff ( ) = a 2 d t = a ) 2 ( T | | T T = otherwise. 222 Example 2 Find the autocorrelation function of the sinusoid f ( t ) = sin( t + ). Since f ( t ) is periodic, the autocorrelation function is defined by the average over one period t + T 1 ff ( ) = f ( t ) f ( t + )d t. T t and with t = 0 2 / ff ( ) = 2 sin( t + ) sin(( t + ) + ) d t 1 = cos( t ) 2 and we see that ff ( ) is periodic with period 2 / and is independent of the phase ....
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lecture_22 - MIT OpenCourseWare http://ocw.mit.edu 2.161...

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