This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 autocorrelation and crosscorrelation functions as measures of self and crosssimilarity 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 ....
View Full
Document
 Fall '08
 DerekRowell

Click to edit the document details