r6_spectrarandom

r6_spectrarandom - 13.42 Design Principles for Ocean...

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13.42 Design Principles for Ocean Vehicles Reading # 13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005 1. STATIONARY AND ERGODIC RANDOM PROCESSES Given the random process y ( z , t ) we assume that the expected value of the random process is zero, however this is not always the case. If the expected value equals some constant x we can adjust the random process such that the expected value is indeed zero: o ( y ( z , t ) = x t , z ) - x . o Again we note that for the stationary ergodic random process the time statistics and event statistics are equal. We write the autocorrelation R () t : 1 T t t { ( y + t , z ) } = R ) = li m yty ( t + t ) dt (1) R = E , z t T fi¥ T 0 i i CORRELATION PROPERTIES 1. R (0) = variance = s 2 = (RMS) 2 0 2. R = R t - t () | 3. R (0 ‡| R t EXAMPLE : Consider the following random process that is a summation of cosines of different frequencies – similar to water waves. N ( )) (2) y ( z , t ) = a cos w t + yz n n n n = 1 where are all independent random variables in [0, 2 p ] with a uniform pdf. This n ©2004, 2005 A. H. Techet 1 Version 3.1, updated 2/24/2005
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13.42 Design Principles for Ocean Vehicles Reading # random process is stationary and ergodic with an expected value of zero. The autocorrelation R () t is N 2 n R t ( ) = a cos wt ) (3) n 2 n = 1 2. SPECTRUM Given a random process that is stationary and ergodic, with an expected value of zero and autocorrelation R t , the power spectral density, or spectrum, of the random process is defined as the Fourier transform of the autocorrelation. ¥ w t - i wt d S = R e t (4) Conversely, the autocorrelation, R t , is the inverse FT of the spectrum 1 ¥ i wt t w R = S ed w (5) 2 p Properties of the Spectrum S w of y ( z , t ) : 1. S w is a real and even function. Since R t is real and even. ¥ ¥ t t } d R e - i d t = R ( ){co s w t - i si n w tt 2. - ¥ 3. It can be shown that the sine component integrates to zero. 4. The variance of the random process can be found from the spectrum: 1 ¥ 2 ww 5. s = ( RM S ) 2 = R (0 = 2 p S d 6. The spectrum is positive always: S )0 w 7. With some restrictions it can also be established that ©2004, 2005 A. H. Techet 2 Version 3.1, updated 2/24/2005
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13.42 Design Principles for Ocean Vehicles Reading # T 1 w S ( ) = li m w ( y t , z ) e - it dt k T fi¥ 2 p - T (Beyond the scope of this course – see Papoulis p. 343 for more info) A spectrum covers the range of frequencies from minus infinity to positive infinity ( < w < +¥ ). + w A one-sided spectrum , S () , is a representation of the entire spectrum only in the positive frequency domain. This one-sided spectrum is convenient and used traditionally, but is not strictly correct. The one sided spectrum is a representation of the entire spectrum only in the positive 1 frequency domain. We “fold" the energy over w = 0 and introduce the 2 p factor we get: 2 w 0 S S + () = 2 p (9) w 0 else This representation for the one-sided spectrum comes from the variance, R (0): 1 ¥ 2 ¥ w = S () d R (0 = s 2 = S d 2 p ww (10) 0 2 p ©2004, 2005 A. H. Techet 3 Version 3.1, updated 2/24/2005
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13.42 Design Principles for Ocean Vehicles Reading #
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.22 taught by Professor Alexandratechet during the Spring '05 term at MIT.

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r6_spectrarandom - 13.42 Design Principles for Ocean...

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