13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 1 Version 3.1, updated 2/24/2005 13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005 1. STATIONARY AND ERGODIC RANDOM PROCESSES Given the random process ()ytζ, 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 ox we can adjust the random process such that the expected value is indeed zero: ()()oytxtxζζ,=,−. Again we note that for the stationary ergodic random process the time statistics and event statistics are equal. We write the autocorrelation ()Rτ: 01(){()()}()lim()()TtiiTREytytRytytdtTτζτζττ→∞=,+,==+∫ (1) CORRELATION PROPERTIES 1. (0)R= variance 2σ== (RMS)20≥ 2. ()()RRττ=− 3. (0)()RRτ≥|| EXAMPLE: Consider the following random process that is a summation of cosines of different frequencies – similar to water waves. 1()cos(())Nnnnnytatζωψζ=,=+∑ (2) where ()nψζ are all independent random variables in [0,2π] with a uniform pdf. This13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 2 Version 3.1, updated 2/24/2005 random process is stationary and ergodic with an expected value of zero. The autocorrelation ()Rτ is 21()cos()2NnnnaRτωτ==∑ (3) 2. SPECTRUM Given a random process that is stationary and ergodic, with an expected value of zero and autocorrelation ()Rτ, the power spectral density, or spectrum, of the random process is defined as the Fourier transform of the autocorrelation. ()()iSRedωτωττ∞−−∞=∫ (4) Conversely, the autocorrelation, ()Rτ, is the inverse FT of the spectrum 1()()2iRSedωττωωπ∞−∞=∫ (5) Properties of the Spectrum ()Sω of ()ytζ,: 1. ()Sω is a real and even function. Since ()Rτ is real and even. 2. ()(){cossin}iRedRidωττττωτωττ∞∞−−∞−∞=−∫∫ 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: 5. 221()(0)()2RMSRSdσωωπ∞−∞===∫ 6. The spectrum is positive always: ()0Sω≥ 7. With some restrictions it can also be established that13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 3 Version 3.1, updated 2/24/2005 1()lim()2TitkTTSytedtωωζπ−−→∞=,∫ (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 (ω−∞<<+∞). A one-sided spectrum, ()Sω+, 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 frequency domain. We “fold" the energy over 0ω= and introduce the 12π factor we get: 2()0()20SSelseωωωπ+≥= (9) This representation for the one-sided spectrum comes from the variance, (0)R: 2012(0)()()22RSdSdσωωωωππ∞∞−∞===∫∫ (10)13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 4 Version 3.1, updated 2/24/2005 which we can rewrite in terms of the one-sided spectrum 20()Sdσωω∞+=∫ (11) where 2()()for02SSωωωπ+=;≥ (12) The spectrum provides a distributed amplitude, or “probability density” of amplitudes, indicating the energy of the system. 3. Application of Spectrum to LTI systems 1. Linear time invariant system We can use the spectrum to help us analyze linear time invariant systems. Since the LTI system is characterized by its impulse response, ()ht, given an input, ()ut, the output can be found from the convolution of the impulse response and the input: ()()()ytutht=∗ (13)13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 5 Version 3.1, updated 2/24/2005 or the Fourier transform of the output is equal to the Transfer function, H(ω), times the FT of the input: ()()()yHuωωω=%% (14) For such a LTI system, if ()ut is a stationary and ergodic random process then ()yt is also stationary and ergodic. Defining the spectrum of ()ut as ()uSω and the spectrum of ()yt as ()ySω we can show that the following holds true: 2()()()yuSHSωωω=|| where 2()Hω is square of the magnitude the transfer function of the LTI system. This is known as the Wiener-Khinchine Relation. We would like to use this relationship and properties of the spectrum to gain insight about the system output, essentially the statistics of the output, knowing only the input and the system transfer function. 4. SHORT TERM STATISTICS Since we are interested in obtaining the statistics associated with the random processes we can use the Spectra to calculate them. As an example, lets look at a spectrum, ()uSω, of sea elevations which consists of many harmonic components. The central limit theorem from probability says, given that there are many events, the sea elevation will have a gaussian distribution. If we assume that the input function, ()ut, is a stationary and ergodic random process with a gaussian pdf, then the the output function, ()yt is also stationary and ergodic with a gaussian pdf. This assumption is good for “short” time intervals, on the order of a storm or an afternoon, but not necessarily over weeks or decades.13.42 Design Principles for Ocean Vehicles Reading # ©2004, 2005 A. H. Techet 6 Version 3.1, updated 2/24/2005 We are interested in gathering the statistics of ()yt given the spectrum ()ySω. The waveheights, ih, and wave periods of interest iT are the random variables in this problem. We have already given that this is a stationary and ergodic random process thus we know that the time statistics are equivalent to the event statistics, we can also show that if ()yt is a realization of the random process ()ytζ, (which is stationary and ergodic) then ergodicity says that ih and iT will provide the statistics on ()ytζ, and vice versa. Often we need to know how often is a certain level is exceeded by the process, in this case the wave height. In order to determine this, we can look at the occurrences of UPCROSSINGS only in a certain time period of interest. (There is further information on this subject in section 3 of the supplemental notes by