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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 autocorrelation and crosscorrelation functions as measures of self and crosssimilarity as a function of delay τ . We continue the discussion here. 1.1 The Autocorrelation Function There are three basic definitions (a) For an infinite duration waveform: T/ 2 1 φ ff ( τ ) = lim f ( t ) f ( t + τ )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 ρ ff ( τ ) = f ( t ) f ( t + τ )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 φ ff ( τ ) = f ( t ) f ( t + τ )d t T t for an arbitrary t , which again may be considered as a “power” based definition. 1 copyright c D.Rowell 2008 22–1 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 ρ ff ( τ ) = f ( t ) f ( t + τ )d t The autocorrelation function is developed graphically below T − τ ρ ff ( τ ) = a 2 d t = a τ ) 2 ( T −   − T ≤ τ ≤ T = otherwise. 22–2 Example 2 Find the autocorrelation function of the sinusoid f ( t ) = sin(Ω t + φ ). Since f ( t ) is periodic, the autocorrelation function is defined by the average over one period t + T 1 φ ff ( τ ) = f ( t ) f ( t + τ )d t. T t and with t = 0 2 π/ Ω Ω φ ff ( τ ) = 2 π sin(Ω t + φ ) sin(Ω( t + τ ) + φ ) d t 1 = cos(Ω t ) 2 and we see that φ ff ( τ ) is periodic with period 2 π/ Ω and is independent of the phase φ ....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.161 taught by Professor Derekrowell during the Fall '08 term at MIT.
 Fall '08
 DerekRowell

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